On The Equivalence Problem for Geometric Structures, I
OON THE EQUIVALENCE PROBLEM FORGEOMETRIC STRUCTURES, I
ANTONIO KUMPERA
Abstract.
We discuss the local and global problems for the equiv-alence of geometric structures of an arbitrary order and, in latersections, attention is given to what really matters, namely theequivalence with respect to transformations belonging to a givenpseudo-group of transformations. We first give attention to gen-eral prolongation spaces and thereafter insert the structures in theirmost appropriate ambient namely, as specific solutions of partialdifferential equations where the equivalence problem is then dis-cussed. In the second part, we discuss applications of all this ab-stract nonsense and take considerable advantage in exploring ÉlieCartan’s magical trump called transformations et prolongementsmériédriques that somehow seem absent in present day geometry. Introduction
Structures have been studied for a long time and they might even beretraced back to Archimedes in his effort to confront and, if possible,to repel the Roman fleet during the siege of Syracuse. The structure ofhis parabolic incendiary mirrors would burn up the roman vessels withtheir beams focused on the sails. Syracuse was, after two long years,finally taken over on a cloudy day by roman general Marcellus andArchimedes killed by a roman soldier. More than two thousand yearslater, Élie Cartan undoubtedly brought the most significant contribu-tions due to his magical skill in dealing with continuous finite and infi-nite groups of transformations ([2]). Our account, given here, is hope-fully a non-tedious though, at best, just a partial repetition of what thereader can certainly find hidden somewhere in Cartan’s
OEuvres ([13]).For this, however, he must not only read french but, above all, have
Medusa ’s penetrating and petrifying stare. A less frightening alterna-tive is offered by Mona Lisa’s regard énigmatique which entails her to
Date : May 2014.2010
Mathematics Subject Classification.
Primary 53C05; Secondary 53C15,53C17.
Key words and phrases. prolongation spaces · structures · equivalence · differen-tial invariants. a r X i v : . [ m a t h . DG ] D ec ANTONIO KUMPERA stare directly into your eyes wherever you should be standing in the
Louvre’s
Main Hall. The reader should also be very careful since, asmany claim, Cartan’s
OEuvres might uncover a Pandora box.The local equivalence problem for structures (or almost-structures)treated systematically and placed in a context of ample generality canbe retraced back to Sophus Lie following the appearance, in the Math-ematische Annalen ([24]), of his renowned mémoire
Über Differential-invarianten (an english translation being available in [1]) and muchlater of his masterpiece
Verwertung des Gruppenbegriffes für Differen-tialgleichungten, I ([25]). Needless to say, a structure for Sophus Lieacquired the more visible nature of a "geometric object" not resemblingat all, at least at first sight, to the crushing formalisms of our presentdays and his theory of differential invariants had the precise aim ofdevising the "best"(shortest, most accurate and of lowest degree) in-tegration methods for differential equations, based on the properties(structure) of the invariance groups. He showed, for example, that Ja-cobi’s last multiplier method was the best possible due to the fact thatthe pseudo-group of all volume preserving transformations is simple!In this restricted context, the structures stand out as solutions of dif-ferential equations ([21]). It should be stated however that whereasLie was incredibly successful in dealing with equations of "finite type"(the solutions depending only upon a finite number of parameters)since he had a deep knowledge and understanding of "finite continu-ous groups", in the general context he was unable to go any furtherbeyond examining a few specific examples involving the simple infinite(transitive) groups. He was aware of all the four classes of (complex)simple groups but rather uneasy whether these were the only ones, thuspreventing him to take any benefits stemming from a systematic use ofJordan-Hölder resolutions ([25]). Consequently, it remained to Cartanto develop the infinite dimensional theory with a touch only accessibleto the most illuminated (see the references [3, 4, 5, 7, 8, 12]).In this paper we concentrate mainly on certain ideas relating the differ-ential invariants of a Lie pseudo-group with the formal and local equiv-alence problem for structures. Apart Lie’s basic ideas, we also find hereCartan and Ehresmann’s most inspiring sources ([6, 9, 15]). To avoidlong repetitions, the technical aspects adopted here stem entirely from[19] and [22] though we try to simplify as far as possible the notationsas well as avoid any excessive abstractions (in Donald Spencer’s words, the abstract nonsense ). Furthermore, perhaps overdue attention hasbeen given to the differential structure of the equations defining k − th N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 3 order equivalences of structures. Whereas in the theory of finite di-mensional Lie groups the three fundamental theorems of Lie shine asneat and as beautiful as Botticelli’s
La Nascita di Venere and Ingre’s
Le Printemps , in the case of groupoids and Lie pseudo-groups theylook more like Picasso’s
Guernica and are a painful headache. In ourcase however, we only have to cope with the second theorem that for-tunately is, in the present situation, very condescending and indulgent(sect.4,5,6, [10], [11]).Last but not least, here are a few words for "peer reviewers". Curi-ously enough, Sophus Lie and Élie Cartan did always row off the "mainstream" for the simple reason that, at their time, essentially nobodywas able to understand their writings. It took Cartan to understandwhat Lie did mean and Charles Ehresmann to understand Cartan.Significantly enough, Lie was Cartan’s thesis adviser and Cartan wasEhresmann’s adviser. As for Sophus Lie, he in fact never needed anyadviser at all since he began writing in Norwegian so nobody would un-derstand him anyway. Unfortunately, the author is unable to pinpointanybody who did or who does (or who ever will) really understand thefull extent of Ehresmann’s thoughts in all their galactic magnitude.Most probably we shall have to await for the next millennium .2. Ehresmann’s prolongation spaces
Let P be a differentiable manifold where, for convenience, we assumeall the data of class C ∞ though it would suffice to assume differentia-bility just up to a certain order. A finite prolongation space of P is aquadruple ( E, π, P, p ) where E is a differentiable manifold called the to-tal space of the prolongation, π : E −→ P a fibration (surmersion) and p a prolongation operator that associates to each local diffeomorphism ϕ of P a local diffeomorphism pϕ of E whose source and target are π -saturated open sub-sets inverse images of the source and target, re-spectively, of ϕ and that furthermore obey the following requirements: i ) pϕ commutes with ϕ and the projection π , ii ) p is local and preserves pastings ( recollements ), Il est à remarquer cependant que Ehresmann, bien qu’il construisait de trèsbeaux arcs-en-ciel, ne s’est jamais soucié d’aller chercher le trésor se trouvant aubout. Par contre, Lie ainsi que Cartan allaient chercher désormais ce trésor sans sesoucier à peindre au préalable de beaux arcs-en-ciel.
ANTONIO KUMPERA iii ) p is a groupoid functor with respect to local diffeomorphisms( ϕ being composable with ψ whenever α ( ψ ) ∩ β ( ϕ ) is non void, theunities being the identities on the open sets), iv ) Every differentiable one-parameter family ( ϕ t ) of local diffeo-morphisms of P prolongs, by p , onto a one-parameter family of localdiffeomorphisms of E for which the vector field d/dt ( pϕ t ) t =0 dependsonly upon d/dt ( ϕ t ) t =0 and projects onto it by T π .We shall say that pϕ is the prolongation of ϕ and, in order to simplifynotations, the prolongation space will just be denoted by E .Much in the same way, an infinitesimal prolongation space of P isa quadruple ( E, π, P, p ) where the prolongation operator p associatesto each local vector field (infinitesimal transformation) ξ given on P , a prolonged vector field pξ defined on the inverse image of the source α ( ξ ) ,this operation satisfying the corresponding (infinitesimal) properties: i ) pξ is π -projectable onto ξ , ii ) p is local and preserves pastings, iii ) p is a pre-sheaf morphism of Lie algebras.Any finite prolongation space determines uniquely an infinitesimal pro-longation space by which it is generated but what really matters is theconverse that is not always true as we shall see in the sequel. Most fibrebundles considered in geometry ( e.g. , tensor bundles, Cartesian framesand co-frames, Stiefel truncated frames and co-frames, Grassmanniancontact elements and their corresponding higher order analogues) areof course finite or infinitesimal prolongation spaces or both though ourmain interest is directed towards jet spaces. Let us also observe thatprolongation spaces "compose" since the prolongation algorithm itselfcan be composed.Let π : P −→ M be a fibration (surmersion), denote by J k P the k − th order jet bundle of local sections of π and α , β , ρ hk the wellknown projections. Following Ehresmann, we also denote by Π k P thegroupoid of all invertible k − jets of the manifold P ( k − jets of localdiffeomorphisms), by J k T P the vector bundle of all k − jets of localsections of T P −→ P i.e. , the jets of local vector fields on P and finallyby ˜ J k T P the fibration of all k − jets of local sections of the compositefibration T P −→ P −→ M . In the sequel, this tilde notation willalways be used for jets of sections of composite fibrations. N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 5
Lemma 1.
Given any fibration π : P −→ M , the jet space J k P has a natural infinitesimal prolongation space structure ( J k P, β, P, p k ) where β is the target map and where p k is the k − th order standardprolongation operator for vector fields (by the target). Though the prolongation morphism p k goes back to Sophus Lie (incoordinates), we would like to add a few words so as to avoid anymisunderstanding. It is not possible to prolong, to J k P , any local dif-feomorphism ϕ of P since such a map can upset transversality (genericposition) of a local section with respect to the π -fibres. However,when this condition is fulfilled, we can transform (at least locally) any π -section, whose image is contained in the domain of ϕ , into a new π -section and thereafter take its k -jet. In particular, we shall thenbe able to prolong any π -projectable local diffeomorphism ϕ of P andthe prolongation functor thus obtained, on projectable maps only, willof course fulfill the above stated properties of a finite prolongationspace. Since we can always define a local vector field by its (local)one-parameter group ( ϕ t ) t and since ϕ = Id , there is no restrictionwhatsoever in the prolongation procedure, to k − th order, of any lo-cal vector field defined on P whereupon J k P becomes an authentic infinitesimal prolongation space of P .Inasmuch, we can say that T P, J k T P, T M and J k T M are prolongationspaces (finite and infinitesimal) of P and M respectively and, moreover,that ˜ J k T P is an infinitesimal prolongation space not only of TP butalso of P for we can first prolong the local vector field ξ , defined on P , to TP and thereafter proceed with the above described "jet spaceprolongation". It should also be noted that ˜ J k T P is a (locally trivial)vector bundle with base space P since the k-th order tangency renderspossible the vector space operations on the fibres. Finally, we would liketo "stress" the condition ( iii ) above by writing explicitly the equality p ( f ξ ) = f pξ where f is any function.As is well known, ( h + k ) − jets can become h − jets of k − jets and, inas-much, ( h + k ) − jets can operate on "split" jets this motivating thefollowing definitions: Definition 1. a) The finite prolongation space E is said to be of order (cid:96) when, for any k ≥ , the k-jet of pϕ at the point z ∈ E only dependsupon the ( (cid:96) + k ) − jet of ϕ at the point y = π ( z ) ∈ P .b) The infinitesimal prolongation space E is said to beof order (cid:96) when, for any k ≥ , the k − jet of pξ , at the point z ∈ E ,only depends upon the ( (cid:96) + k ) − jet of ξ at the point y = π ( z ) ∈ P . ANTONIO KUMPERA
Recalling that the jet bundle J k T M identifies with the Lie algebroid ofthe Lie groupoid Π k M , we can define for the above prolongation spacesof finite order and for any fixed positive integer k :a) A left action(1) Λ k : Π (cid:96) + k P × P Π k E −→ Π k E of the Lie groupoid Π (cid:96) + k P on the groupoid Π k E (one can actuallyreplace the last groupoid by the space of all k − jets J k ( E, E ) ) by setting(2) ( j (cid:96) + k ϕ ( βX ) , X ) (cid:55)−→ j k ( pϕ )( βX ) · X , the fibre product being taken with respect to α , for the first factor,and with respect to π ◦ β , for the second factor,b) its infinitesimal generator namely, the morphism(3) λ k : J (cid:96) + k T P × P E −→ J k T E defined by(4) ( j (cid:96) + k ξ ( y ) , X ) (cid:55)−→ j k ( pξ )( X ) , y = πX , c) and, finally, its extension to an infinitesimal action on the jets ofthe tangent bundles(5) T Λ k : J (cid:96) + k +1 T P × P J k +1 T E −→ J k T E defined by the left Lie bracket action(6) ( j (cid:96) + k +1 ξ ( y ) , X ) (cid:55)−→ [ j k +1 ( pξ )( αX ) , X ] , y = παX , where we are forced to augment the order by 1 since brackets absorbone order of differentiation. The reason for putting in evidence the Liebracket becomes apparent if we operate, as is usually done, by takinglocal one parameter groups and thereafter differentiating.We now claim that the action Λ k is differentiable. In fact, using stan-dard methods involving the Lie algebroid J k T P of the groupoid Π k P (or, if one prefers, the sheaf J k T P ), we can define a local exponentialmap that, with the help of the property ( iv ) will provide the requireddifferentiability. It then also follows that the infinitesimal generator λ k N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 7 as well as the infinitesimal action T Λ k are differentiable though thisproperty can be proved directly by observing that the vector bundlesinvolved are locally trivial and generated by local holonomic sections.It should also be observed that T Λ k is bilinear over R when the sourceand target spaces are fibered over E .We next observe that given a finite number of "composable" prolon-gation spaces, each of finite order, the composed prolongation spaceis also of finite order equal to the sum of the individually prescribedorders.Other prolongation spaces that will be of our interest are those de-scribed in the following Lemma 2.
To each finite or infinitesimal prolongation space ( E, π, P, p ) and to each positive integer h corresponds, in a canonical way, a pro-longation space ( J h E, α, P, p h ) verifying the following properties:a) When E is of finite order (cid:96) then J h E is of order (cid:96) + h .b) The target projection β : J h E (cid:55)−→ E is a surjective morphism ofprolongation spaces i.e., respects the fibrations over P and commuteswith the respective prolongation operations.c) More generally, the projection ρ k,h : J h E −→ J k E is a prolonga-tion spaces morphism. Here, J h E is the set of all h -jets of local sections of π and is calledthe standard h-th order prolongation of E , the prolongation operationbeing the composite of the operation provided by E followed by thestandard jet prolongation. Needless to say, everything that was statedconcerning the prolongation space J k P in the Lemma 1 can be para-phrased ipsis litteris for the above data. Inasmuch, we can also repeateverything that was said previously for the prolongation space ˜ J k T P of infinitesimal variations relative to the composite T P −→ P −→ M , the second arrow being equal to π , as well as for the k − th order varia-tions space ˜ J k E composed of all k -jets of local sections of the compositefibration E −→ P −→ M , where E −→ P is a prolongation space and P −→ M simply a fibration giving rise to the finite or infinitesimalvariations ( cf. , [20] for the definitions). As for the bracket operationconsidered previously when defining an infinitesimal action, it is welldefined in the present context due to the contact order conditions im-posed. Finally, all the above considerations also extend to pre-sheaves ANTONIO KUMPERA
Γ( ) of local sections and enable us to operate with locally defined ob-jects. In the sequel we shall also need the following extension of (3)namely,(7) λ k : J (cid:96) + k T P × P J k E −→ T J k E , defined by ( j (cid:96) + k ξ, j k σ ) = p k ◦ p ( ξ )( j k σ ) and where T P −→ P is thetangent prolongation space. In much the same way and using theprolongation operator, we can define the morphism(8) ˜ λ k : J (cid:96) + k T P × P J k E −→ J k T E , as well as the semi-holonomic extension(9) λ k + h : J (cid:96) + k + h T P × P J k E −→ J h ( T J k E ) . We thus see that the choices are many and, in fact, we could go on muchfurther with the Ehresmannian game of the jeu de la théorie des jets by considering for instance semi-holonomic, sesqui-holonomic and (def-initely) non-holonomic jets but fortunately these will be of no purposeto us so we might as well forget about them right away. Furthermore,and this will be very useful, we can play the Ehresmannian game with differential forms and co-tangent bundles that, in this case, will act co-variantly . As for the prolongation operation, there is of course es-sentially just one such operation that can however be vested under twoor three garbs. Instead of prolonging by the target, as is done in thepresent paper, we can also prolong by the source or even, combiningthe two, we can prolong via the anchor , the same one that holds Kirillanchored and not on the sail. In later sections we shall also introduce merihedric prolongations ( prolongements mériédriques de Élie Cartan )as well as such transformations on jet spaces and higher order Grass-mannians. Contrary to what is standard, there are uncountably manypossibilities for the merihedric functor each one having its own mer-its and outstanding performance. In fact, Medusa as well as MonaLisa claim that the merihedric setup was the magical trump and jokerhidden in Cartan’s sleave. Fortunately we shall need not talk aboutmerihedric jet spaces, the standard ones being still of good use. Onelast remark: When differentiability is replaced by analyticity in theinitial requirements for the prolongation spaces then, of course, all theother data also become analytic. We hope as well that the reader al-ready noticed our small notational changes. Instead of the standard
N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 9 j kx σ , we prefer to write j k σ ( x ) and, instead of J k , we write J k , suchnotations rendering more pleasant and "co-variant" their composites.3. Symbols
In this section we shall often refer to former publications ([19], [22])for the notations and the results which, however, are completely stan-dard and well known. In this sense, we denote by D the Spenceroperator, by S k the bundle of symmetric tensors and by δ the Spenceroperator restricted to the principal parts. Our first task is to examinewhat happens with the kernels and consequently write: −→ S (cid:96) + k T ∗ P ⊗ T P × P E −→ J (cid:96) + k T P × P E −→ J (cid:96) + k − T P × P E −→ ↓ (cid:96) k ↓ λ k ↓ λ k − −→ S k T ∗ E ⊗ T E −→ J k T E −→ J k − T E −→ as well as → S (cid:96) + k T ∗ P ⊗ T P × P J k E → J (cid:96) + k T P × P J k E → J (cid:96) + k − T P × P J k − E → ↓ (cid:96) k ↓ λ k ↓ λ k − → S k T ∗ E ⊗ V E × E J k E −→ T J k E −→ T J k − E × J k − E J k E → and observe that the above λ k is equal to λ relative to the prolongationspace J k E of P , the later being of finite order (cid:96) + k . More generally andwith the obvious notations, λ k + h ( E ) = λ h ( J k E ) . Next, we claim thateach family ( (cid:96) k ) and ( (cid:96) k ) is a natural transformation of the correspond-ing δ − cohomology complexes this becoming apparent by examiningthe commutativity of the two diagrams below and observing that thevertical map (cid:96) k in the second diagram is in fact equal to [ S (cid:96) + k T ∗ P ⊗ T P × P J E ] × J E J k E (cid:96) k × Id −−−→ [ S k T ∗ E ⊗ V E × E J E ] × J E J k E the term (cid:96) k depending only upon its projection in J E . In order toshow the naturality we are forced, of course, to confront the abovemaps with the Spencer operator hence the necessity in extending theactions to the sheaf level.We first consider the diagram (10), extend the infinitesimal generator(3) to the pre-sheaf of local sections and derive thereafter the sheafmorphism − → S (cid:96) + k T ∗ P ⊗ T P × P E δ × I d −−− → T ∗ P ⊗ S (cid:96) + k − T ∗ P ⊗ T P × P E δ × I d −−− → ∧ T ∗ P ⊗ S (cid:96) + k − T ∗ P ⊗ T P × P E δ × I d −−− → ··· ( ) ↓ (cid:96) k ↓ π ∗ ⊗ (cid:96) k − ↓ π ∗ ⊗ (cid:96) k − − → S k T ∗ E ⊗ T E δ − → T ∗ E ⊗ S k − T ∗ E ⊗ T E δ − → ∧ T ∗ E ⊗ S k − T ∗ E ⊗ T E δ − → ··· → S (cid:96) + k T ∗ P ⊗ T P × P J E δ × I d −−− → T ∗ P ⊗ S (cid:96) + k − T ∗ P ⊗ T P × P J E δ × I d −−− → ∧ T ∗ P ⊗ S (cid:96) + k − T ∗ P ⊗ T P × P J E δ × I d −−− → ··· ( ) ↓ (cid:96) k ↓ I d T ∗ P ⊗ (cid:96) k − ↓ I d ∧ T ∗ P ⊗ (cid:96) k − → S k T ∗ E ⊗ V E × E J E δ × I d −−− → T ∗ P ⊗ S k − T ∗ E ⊗ V E × E J E δ × I d −−− → ∧ T ∗ P ⊗ S k − T ∗ E ⊗ V E × E J E δ × I d −−− → ··· N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 11 λ k : J (cid:96) + k T P × P E −→ J k T E, where λ k ( σ y , q ) = (Γ( λ k ) σ ) q , y = π ( q ) , Γ( ) denoting the set of alllocal sections. It now suffices to show that λ k commutes with D or, inother words, that the diagram (12) below commutes: J (cid:96) + k T P × P E D × Id −−−−→ ( T ∗ P × P E ) ⊗ ( J (cid:96) + k − T P × P E ) (12) ↓ λ k ↓ π ∗ ⊗ λ k − J k T E D −→ T ∗ E ⊗ J k − T E
The commutativity is obvious on holonomic sections since D vanisheson them and thereafter it suffices to argue as in [22], pg.75, using theformula (2) shown on that page. The restriction to the symbols thenproves our claim for the first square of (10). To complete the proof,it suffices to adapt the diagram (12) to the other squares of (10) byiterating successively the Spencer operator. However, the previouslymentioned formula (2) implies immediately the desired result as soonas the first square commutes. The commutativity of (11) is provedessentially in the same manner by observing that λ k is the pullback of λ k via the composite p k ◦ (cid:93) , where (cid:93) : J k T E × E J k E −→ ˜ J k T E is the fibrebundle morphism considered in [19], Propos.11.1 ( M standing in theplace of P ). The proof can then be achieved with the help of the reducedform of the holonomic prolongation operator acting on infinitesimalvariations and the Theorem 12.1.Since the ( (cid:96) k ) and ( (cid:96) k ) are natural transformations of the δ complexesand since the ( λ k ) and ( λ k ) commute with D it also follows that thesenatural transformations are compatible (commute) with the algebraicprolongations operating on the principal parts (i.e., on the symbols).For the sake of not omitting any useful information, let us finally ob-serve that the total spaces of the finite prolongation spaces are alwayslocally trivial bundles over their base spaces, a fact that is not trueanymore for infinitesimal prolongation spaces ( e.g. , withdraw a pointfrom the total space). However, when the fibres over the base spaceare compact and connected then these total spaces also become locallytrivial, this last claim being just a special case of very general resultsdue to Ehresmann. Lie and Cartan’s notion of structure
Let ( E, π, P, p ) be a finite or infinitesimal prolongation space. Definition 2.
A finite or infinitesimal (almost-)structure of species Eon the manifold P is the data provided by a global section (continuous,differentiable, analytic) of the prolongation space E.
We can define as well a structure of species E above an open set U of P by simply taking a local section and thereafter derive the notions of germ of structure, k − jet of structure, k − th order contact element of astructure inasmuch as an infinite jet of a structure (formal structure ata point). We should also mention that the above definitions are not anyof Spencer’s abstract nonsense since many of the well known structures( e.g. , Riemannian, conformal, almost-complex, almost-symplectic, etc.)are defined by global sections of locally trivial tensor bundles withfibres homogeneous spaces of linear groups. Inasmuch, almost-product,contact, Stiefel and many other structures are defined by sections ofwell known bundles in homogeneous spaces. Moreover, as we shall seefurther, the integrability of an almost-structures can be detected bya Pfaffian system (also a structure!) defined on E or on one of itsprolongation spaces.Given a structure S of species E (a finite prolongation space), asso-ciated to it are the pseudo-group Γ( S ) of all its local automorphismsas well as the infinitesimal pseudo-algebra L ( S ) of all its local infini-tesimal automorphisms (vector fields). By definition, ϕ ∈ Γ( S ) when pϕ leaves invariant the sub-manifold im S or, in other terms, when ϕ ( S ) def = pϕ ◦ S ◦ ϕ − = S in the appropriate domains. Much in thesame way, ξ ∈ L ( S ) when ξ generates a local one-parameter group ( ϕ t ) with all its elements in Γ( S ) which amounts to say that pξ is tangentto im S or equivalently that the Lie derivative θ ( pξ ) S = ddt ϕ − t ( S ) | t =0 vanishes (for each fixed Y ∈ im S , take the tangent vector to the curveissued from Y ). Quite often, it is convenient to consider (differentiable)one-parameter families of local transformations that are not necessar-ily local one-parameter groups. With that in mind, let us next remarkthat such a local one-parameter family ( ϕ t ) defined in P is entirelycomposed of elements belonging to Γ( S ) if and only if the followingtwo properties are fulfilled:a) ϕ t ∈ Γ( S ) for some value t ( t varies in an open interval) and Il serait malheureux de l’appeler une structure locale car cette terminologie, dueà Ehresmann, a un sens très précis autre que ci-dessus. Toute structure d’espèce E est une espèce de structure locale. N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 13 b) for each u , T ϕ − u ◦ ddt ( ϕ t ) | t = u ◦ ϕ u ∈ L ( S ) .The set of all local transformations arising from such one-parameterfamilies is obviously equal to Γ( S ) . When E is just an infinitesimalprolongation space, we can still define L ( S ) by using the tangency withrespect to im S and afterwards defining Γ inf ( S ) as being the pseudo-group generated (through composition and gluing) by the set of all theelements belonging to the local one-parameter families verifying ϕ t = Id together with the above condition (b). We can further simplify (re-parametrize) by taking t = 0 and clearly Γ inf ( S ) ⊂ Γ( S ) when E isalso a finite prolongation space, L ( S ) being its infinitesimal generator.The condition ϕ t = Id is an acceptable simplification (by composing,if necessary, with ϕ − t ) and enables us to initiate the one-parameterfamily with the automorphism Id that can always be prolonged.Let us now examine the defining equations for Γ( S ) and L ( S ) henceexamine the nature of the infinitesimal jets that are automorphisms of S up to a certain order and this brings us to examine contact prop-erties. In general, being given a differentiable manifold M and twosub-manifolds N, N (cid:48) of the same dimension p and meeting at a point x , we shall say, according to the general definition of higher order Grass-mannians, that these two sub-manifold have a k − th order contact atthe point x when there exists an invertible k − jet µ of N onto N (cid:48) withsource and target x such that j k ι ( x ) = j k ι (cid:48) ( x ) · µ , where ι and ι (cid:48) arethe inclusions of the sub-manifolds. The equivalence classes under thisrelation are called the k − th order contact elements of dimension p onthe manifold M and at the point x . Let us now take a vector field ξ defined in a neighborhood of x . The composite ξ ◦ ι is a local sectionof the vector bundle T M | N −→ N , hence its k − jet at the point x isan element of J k ( T M | N ) and, of course, J k T N ⊂ J k ( T M | N ) . We shallsay that ξ is tangent of order k to N at x when j k ( ξ ◦ ι )( x ) ∈ J k T N .Furthermore, if N and N (cid:48) have a contact of order k + 1 at the point x then T N and
T N (cid:48) have, along any point of T x N = T x N (cid:48) a contactof order k when considered as sub-manifolds of T M : The contact re-lation is obtained by extending to tangent vectors the initial contactrelation. In particular, we infer that the vector fields tangent of order k to the sub-manifold N at x are the same as those tangent to N (cid:48) .Conversely, the last condition (on the tangency of T N with
T N (cid:48) ) im-plies the ( k + 1) − st order tangency of N with N (cid:48) at the point x ([30],Propos.3, pg.20). Observing that j k ( ξ ◦ ι )( x ) = j k ξ ( x ) · j k ι ( x ) , we con-clude that the k − th order tangency concept for vector fields is more properly a notion of tangency of the elements of J k T M with a sub-manifold N or, in a more general form, with the elements belonging tothe fibre bundle of ( k + 1) − st order p − dimensional contact elements of M ( p = dim N ). We finally remark that the groupoid Π k M operatesto the left on the fibre bundle of k − th order contact elements. Thereis an alternative definition, due to Ehresmann, stating that a k − th order contact element is a linear subspace in the space of k − th ordertangent vectors. Though apparently simpler, this definitions seems tolack visibility. It is easy to see tangency but not so easy to see partialderivatives.We now translate the above definitions of contact order into a formmore suitable to our context, return to the local automorphisms andconsider, to start with, a finite prolongation space E −→ P of order (cid:96) as well as a structure S of species E . A local diffeomorphism ϕ of P is called a k − th order automorphism of S at the point y wheneverthe image sub-manifold pϕ ( im S ) has a k − th order contact with imS at the point S ◦ ϕ ( y ) . Since the contact only depends on j k pϕ ( S ( y )) and the prolongation space is of finite order (cid:96) , we infer that this notionboils down to the following: An element Z ∈ Π (cid:96) + k P with source y andtarget y’ is a k − th order automorphism of S when pZ , k − jet of source S ( y ) corresponding to Z ( ( (cid:96) + k ) − jets prolong to k − jets), transformsthe k − th order contact element of S at S ( y ) onto the k − th ordercontact element of S at S ( y (cid:48) ) . However, the contact element pZ ( S ) isrepresented by the image of the local section pϕ ◦ S ◦ ϕ − where ϕ is arepresentative of Z i.e., Z = j (cid:96) + k ϕ . Moreover, the images of two localsections σ and σ (cid:48) of E define the same k − th order contact element ata common point z ∈ E if and only if j k σ ( πz ) = j k σ (cid:48) ( πz ) and therefore pZ ( S ) and S define the same k − th order contact element at S ( y (cid:48) ) ∈ E if and only if j k ( pϕ ◦ S ◦ ϕ − )( y (cid:48) ) = j k S ( y (cid:48) ) that is to say, when ( pZ ) · j k S ( y ) · Z − k = j k S ( y (cid:48) ) , Z k = ρ k,(cid:96) + k Z. Lemma 3.
A jet Z ∈ Π (cid:96) + k P is a k − th order automorphism of S if andonly if Z ( j k S ( αZ )) = j k S ( βZ ) by means of the left action of Π (cid:96) + k P on J k E (notations: Z ( ) = p k ϕ ( ) , ϕ represents Z and p k = p k ◦ p ).The ( (cid:96) + k ) − th order equation defining Γ( S ) is provided by the closedsub-groupoid (P being replaced by α ( S ) whenever S is not global) (13) R (cid:96) + k ( S ) = { Z ∈ Π (cid:96) + k P | Z ( j k S ( αZ )) = j k S ( βZ ) } . N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 15
This equation is not complete in general i.e. , Γ( S ) might be strictlycontained in Sol R (cid:96) + k ( S ) and, furthermore, the inclusion R (cid:96) + k ( S ) ⊃ J (cid:96) + k Γ( S ) might also be strict.We next consider the non-linear differential operator D ( S ) : ϕ ∈ Diff P (cid:55)−→ ϕ − ( S ) = ( pϕ − ) ◦ S ◦ ϕ ∈ E that is of order (cid:96) since it is obtained by the composition of j (cid:96) and thefibered morphism Φ( S ) : Y ∈ Π (cid:96) P (cid:55)−→ Y − ( S ( βY )) ∈ E where Y ( ) is the action of the groupoid Π (cid:96) P on E . The k − th orderprolongation of D ( S ) is the operator of order (cid:96) + kp k D ( S ) : ϕ ∈ Diff P (cid:55)−→ j k ( pϕ − ◦ S ◦ ϕ ) = p k ϕ − ◦ j k S ◦ ϕ == ϕ − ( j k S ) ∈ J k E and is obtained, with the aid of j (cid:96) + k , by means of the fibered morphism p k Φ( S ) : Z ∈ Π (cid:96) + k P (cid:55)−→ Z − ( j k S ( βZ ) ∈ J k E where Z ( ) is the action of Π (cid:96) + k P on J k E . The following short non-linear sequences −→ R (cid:96) + k ( S ) −→ Π (cid:96) + k P p k Φ( S ) −−−−→ ( J k E, j k S ) and −→ Γ( S ) −→ Diff P p k D ( S ) −−−−→ ( J k E, j k S ) are therefore exact in the set theoretical sense. When im p k Φ( S ) is asub-manifold of J k E and p k Φ( S ) : Π (cid:96) + k P −→ im p k Φ( S ) is a submersion then R (cid:96) + k ( S ) is a regularly embedded sub-manifold of Π (cid:96) + k P and, for all h , the equation R (cid:96) + k + h ( S ) is the prolongation in theusual sense i.e., the sub-set Π (cid:96) + k + h P ∩ J h R (cid:96) + k ( S ) of R (cid:96) + k ( S ) . Takinginto account the Proposition 2.1 in [16], we are tempted to replace theabove two hypotheses by the unique assumption: p k Φ( S ) : Π (cid:96) + k P −→ J k E is locally of constant rank in a neighborhood of each point belongingto R (cid:96) + k ( S ) . Unfortunately (or perhaps fortunately) the above Propo-sition is inexact. If, with the notations of the above citation, X isreduced to a point, this Proposition would imply the remarkable state-ment: Théorème. Toute immersion est un plongement . We shall returnlater to this matter and show that, in the specific case of the equations R (cid:96) + k ( S ) , the first hypothesis can in fact be replaced by weaker condi-tions.Let us now examine the infinitesimal aspects. A local vector field ξ defined on P is said to be a k − th order infinitesimal automorphismof the structure S at the point y when the prolonged vector field pξ istangent up to order k to the sub-manifold im S at the point z = S ( y ) .Since π ◦ S = Id , this condition can be replaced by j k ( pξ ◦ S )( y ) = j k ( T S ◦ ξ )( y ) , this last condition measuring the "distancing" of pξ from S ∗ ξ along im S in the vicinity of the point z = S ( y ) . The prolongation space E being of finite order (cid:96) , this last condition translates by the following:An element Y ∈ J (cid:96) + k T P of source y is a k − th order infinitesimalautomorphism of S whenever pY , k − jet of source S ( y ) correspondingto Y , is tangent of order k to im S at the point S ( y ) . We next observethat both pξ ◦ S and T S ◦ ξ are local sections of the tangent Lie fibration T E −→ P composite of the natural projection T E −→ E with π and,consequently, the reduced form of the holonomic prolongation ([19],Théorème 12.1) shows that the vector ( p k ξ ) j k S ( y ) ( p k = p k ◦ p ) onlydepends on j k ( pξ ◦ S )( y ) and the second line of the diagram (22.29) inthe above citation, §
22, restricted to the fixed section j k S (the sectionbeing fixed, we are not forced any more to ascend to J k +1 E ), showsthat the jets j k ( T S ◦ ξ )( y ) are precisely those, among the elements of ˜ J k T E , for which the vector associated by prolongation is tangent to im j k S at the point j k S ( y ) . Summing up, we obtain the following Lemma 4.
The jet Y ∈ J (cid:96) + k T P is a k − th order infinitesimal auto-morphism of S if and only if the vector p k Y corresponding to it, at thepoint j k S ( αY ) , is tangent to im j k S ( p k Y = ( p k ξ ) j k S ( αY ) where ξ is arepresentative of Y). Moreover, the equation of order (cid:96) + k of the Liealgebroid L ( S ) is given by N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 17 (14) R (cid:96) + k ( S ) = { Y ∈ J (cid:96) + k T P | p k Y ∈ ( j k S ) ∗ T αY P } . Since the map λ k is a differentiable vector bundle morphism and theassociated infinitesimal action Γ( λ k ) is also a morphism of Lie algebrapre-sheafs, it follows that the linear equation R (cid:96) + k ( S ) is a Lie equa-tion ([22], [27]). Moreover, L ( S ) = Sol R (cid:96) + k ( S ) though, in general,this equation is not complete . We next consider the linear differentialoperator D ( S ) : ξ ∈ T P (cid:55)−→ θ ( ξ ) S = ddt ϕ − t ( S ) | t =0 ∈ V E | im S where V E is the vector sub-bundle of
T E composed by the π − verticalvectors and ϕ t is a local one-parameter family e.g. , the local group,such that ddt ϕ t | t =0 = ξ . The restriction V E | imS can be considereda bundle with base space P since it can be identified to S − V E and,according to the general definitions, D ( S ) is the linearization of D ( S ) along the section Id of Diff P . It is a linear operator of order (cid:96) whoseassociated linear morphism Ψ( S ) : J (cid:96) T P −→ V E | im S is defined as follows: Let Y = j (cid:96) ξ ( y ) , ( ϕ t ) a local one-parameter groupassociated to ξ and Y t = j (cid:96) ϕ t ( y ) . Then Ψ( S )( Y ) = ddt [ Y − t ( Sy t )] t =0 , where y t = βY t and where, for simplicity, we write Sy t though meaning S ( y t ) . However, Y − t ( Sy t ) = pϕ − t ( Sy t ) = pϕ − t ( Sy t ) and, consequently, Ψ( S )( X ) = − ( pξ ) S ( y ) + S ∗ ( ξ y ) . We infer that(15) Ψ( S )( Y ) = − v ( pξ S ( y ) ) , where v ( ) is the vertical component of a vector following the directsum decomposition T S ( y ) E = V S ( y ) E ⊕ S ∗ ( T y P ) along im S .Let us now return to the general definitions of [20], I, §
17 and, in par-ticular, to the last sequence, on pg.341, exhibiting the linear morphism τ k ( T D ) that defines the differential operator T D . In the present case(and adapting slightly the notations), it concerns the vertical lineariza-tion along the section Id for the specific fibration P = M × M −→ M , ( x, x (cid:48) ) (cid:55)→ x , where J (cid:96) P is replaced by J (cid:96) M = J (cid:96) ( M, M ) (each section of P identifies with a map M −→ M ), ˜ J (cid:96) V P | J (cid:96) M = J (cid:96) M × M J (cid:96) T M (fibre product with respect to the projection β ) andwhere p (cid:96) : J (cid:96) M × M J (cid:96) T M −→ V J (cid:96) M simply becomes the canonicalidentification. The morphism T τ (cid:96) ( D ) = T Φ( S ) : V J (cid:96) M | im Id −→ V E | im S becomes, by means of the canonical identification, equal to Ψ( S ) . The k − th order prolongation of D ( S ) is then the linear operator of order (cid:96) + k p k D ( S ) : ξ ∈ T M (cid:55)−→ j k [ θ ( ξ ) S ] ∈ J k ( V E | im S ) , that we can, by means of the isomorphism p k : ˜ J k V E −→ V J k E ([20], Propos.12.2), replace by p k ◦ p k D ( S ) : T M −→ V J k E | im j k S that, in turn, is nothing else but the linearized, along the identitysection, of p k D ( S ) ([20], pg.342). Furthermore, the linear morphismassociated to p k ◦ p k D ( S ) is on the one hand equal to p k ◦ p k Ψ( S ) and on the other, for being a linearization and due to the canonicalidentification, equal to(16) T τ (cid:96) + k ( p k D ) = T p k Φ( S ) : V J (cid:96) + k M | im Id −→ V J k E | im S. An entirely similar calculation, where we shall replace ϕ − t ( S ) = D ( S ) ϕ t by ϕ − t ( j k S ) = p k D ( S ) ϕ t , will show that(17) p k ◦ p k Ψ( S )( X ) = − v ( p k ξ j k S ( x,x (cid:48) ) ) where X ∈ J (cid:96) + k T M and v is the vertical component of a vector inthe direct sum decomposition T Z J k E = V Z J k E ⊕ ( j k S ) ∗ ( T x M ) , Z = j k S ( x, x (cid:48) ) and this implies the exactness of the sequence(18) −→ R (cid:96) + k ( S ) −→ J (cid:96) + k T M p k Ψ( S ) −−−−→ J k ( V E | im S ) for all k ≥ , hence R (cid:96) + k + h ( S ) is the prolongation of order h of theequation R (cid:96) + k ( S ) whenever p k Ψ( S ) is locally of constant rank as a N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 19 linear mapping on each fibre ( i.e. , when R (cid:96) + k ( S ) is a locally trivialvector bundle). Moreover, the sequence(19) −→ L ( S ) −→ T M p k D ( S ) −−−−→ J k ( V E | im S ) is also exact for all k ≥ .5. Local and infinitesimal equivalences
In this section we study the consequences of the previous hypothesesand conditions in view of showing that the micro-differentiable struc-tures introduced by Pradines ([32]) and hereafter considered, do gen-erate the desired global structures. By the canonical identification Π (cid:96) + k P × P J (cid:96) + k T P −→ V Π (cid:96) + k P resulting from the prolongation, bythe target, of local vector fields defined on P , each sub-space R (cid:96) + k ( S ) y determines, at every point X ∈ Π (cid:96) + k P with β ( X ) = y , a sub-space (∆ (cid:96) + k ) X ⊂ V X Π (cid:96) + k P of the same dimension as that of R (cid:96) + k ( S ) andthus defines an α − vertical distribution on Π (cid:96) + k P (field of contact ele-ments) whose point-wise dimension is locally constant if and only if thefibre bundle R (cid:96) + k ( S ) is locally trivial. This last condition being satis-fied, the distribution ∆ (cid:96) + k is integrable since R (cid:96) + k ( S ) is a Lie equation. Proposition 1.
Let S be a structure of species E and finite order (cid:96) .Then ker T p k Φ( S ) = ∆ (cid:96) + k ( S ) and, furthermore, the following proper-ties are equivalent:i) p k Ψ( S ) is locally of constant rank,ii) p k Φ( S ) is locally of constant rank along the units of Π (cid:96) + k P ,iii) p k Φ( S ) is locally of constant rank,iv) ∆ (cid:96) + k is locally of constant dimension.These equivalent conditions being verified, R (cid:96) + k ( S ) is a union of inte-gral leaves of the distribution ∆ (cid:96) + k . Proof . We first observe that ∆ (cid:96) + k is invariant by all the right transla-tions of the groupoid Π (cid:96) + k P , such a translation mapping α − fibres onto α − fibres and preserving the targets. On the other hand, the morphism p k Φ( S ) is a differential co-variant relative to the right action of thegroupoid Π (cid:96) + k P on itself i.e. , the following formula holds:(20) p k Φ( S )( X · Y ) = Y − p k Φ( S )( X ) . We next observe that ker T p k Φ( S ) | im Id = ∆ (cid:96) + k | im Id since p k ◦ p k Ψ( S ) identifies, by means of the canonical identification, to T p k Φ( S ) | im Id = ∆ (cid:96) + k | im Id (cf.(14)) and that the kernel of T p k Φ( S ) : T Π (cid:96) + k P −→ T J k E is equal to that of the restriction T p k Φ( S ) : V Π (cid:96) + k P −→ V J k E . The invariance of ∆ (cid:96) + k and theco-variance of p k Φ( S ) entails the equality everywhere and we infer theequivalence of the four stated properties. If F is a leaf of ∆ (cid:96) + k , then T p k Φ( S ) | T F = 0 and consequently p k Φ( S )( F ) reduces to a point.In particular, when X ∈ F and p k Φ( X ) = j k S ( x ) , x = α ( X ) , then p k Φ( S )( F ) = j k S ( x ) where after F ⊂ R (cid:96) + k ( S ) and this achieves theproof.Let us now observe that, for any fibration morphism λ , P λ −−→ P ↓ ↓ M Id −−→ M the following two conditions are equivalent: a) λ is locally of constant rank,b) λ is locally of vertical constant rank i.e., the rank of the restrictionsof λ to the fibres is locally constant with respect to the topology of P(and not only of the fibres), since we always have the relation rank y λ = dim M +( vertical rank ) y λ .This amounts to say that the rank of T λ : T P −→ T P (cid:48) differs, at eachpoint, from the rank of
T λ : V P −→ V P (cid:48) by the integer dim M and,in particular, that ker T λ = ker T λ | V P . We also observe that theabove properties still hold when we replace Id by any diffeomorphism ϕ of M (and even by a diffeomorphism M −→ M (cid:48) ). In particular, when P and P’ are fibre bundles and λ is a morphism of such bundles, thenthe vertical rank (rank of T λ restricted to the tangent space of a fibre)is equal to the rank of the restriction of λ to the fibres.On the other hand, and without any regularity hypotheses pendingupon p k Φ( S ) or p k Ψ( S ) , we remark that the isotropy(21) ( R (cid:96) + k S ) y = { X ∈ R (cid:96) + k ( S ) | α ( X ) = β ( X ) = y } of R (cid:96) + k ( S ) at the point y is always a closed Lie sub-group of thetotal isotropy (Π (cid:96) + k M ) y since it is given by the "closed" conditions N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 21 X ( j k S ( y )) = j k S ( y ) . Its Lie algebra identifies canonically with thelinear isotropy(22) ( R (cid:96) + k S ) y = { X ∈ R (cid:96) + k ( S ) | α ( X ) = y , β ( X ) = 0 } of R (cid:96) + k ( S ) that on its turn is determined by the condition p k X = 0 .Furthermore, these two remarks show that the finite and infinitesimal k − th order isotropies of S at the point y are entirely determined bythe jet j k S ( y ) . In particular, the jet of order (cid:96) only depends on thepoint S ( y ) ∈ E . Let us denote by R (cid:96) + k ( S ) y the fibre, with respect to α and above the point y , of R (cid:96) + k ( S ) and by B (cid:96) + k ( S ) y its projection β ( R (cid:96) + k ( S ) y ) . Corollary 1. If R (cid:96) + k ( S ) is a locally trivial vector bundle then, at anypoint y, the fibre R (cid:96) + k ( S ) y is a closed and regularly embedded sub-manifold of (Π (cid:96) + k P ) y whose connected components are integral leavesof ∆ (cid:96) + k . Furthermore, B (cid:96) + k ( S ) y is a closed and regularly embeddedsub-manifold of P and the triple ( R (cid:96) + k ( S ) y , β , B (cid:96) + k ( S ) y ) is a locallytrivial sub-fibre bundle of (Π (cid:96) + k P ) y | B (cid:96) + k ( S ) y with structure groupequal to R (cid:96) + k ( S ) y (it being understood that α ( S ) = P ). Proof . It is clear that R (cid:96) + k ( S ) y is closed being the inverse image ofthe point j k S ( y ) relative to the map p k Φ( S ) y : (Π (cid:96) + k P ) y −→ ( J k E ) y .Inasmuch, it is also clear that R (cid:96) + k ( S ) y is a principal space of theisotropy group R (cid:96) + k ( S ) y , the orbits being the inverse images, by β , ofthe points of B (cid:96) + k ( S ) y . The hypothesis on R (cid:96) + k ( S ) entails, in virtueof the previous proposition, that p k Φ( S ) y : (Π (cid:96) + k P ) y −→ ( J k E ) y has a locally constant rank and consequently R (cid:96) + k ( S ) y is a regularlyembedded sub-manifold. Since ker T p k Φ( S ) = ∆ (cid:96) + k ( S ) , we see at oncethat T R (cid:96) + k ( S ) y = ∆ (cid:96) + k ( S ) | R (cid:96) + k ( S ) y and thus infer that the leaves of ∆ (cid:96) + k ( S ) contained in R (cid:96) + k ( S ) y are open sets hence, due to the connex-ity of the leaves, are necessarily the connected components. We denoteby Ξ the distribution defined on P by Ξ y = β ( R (cid:96) + k ( S ) y ) . The rightinvariance of the distribution ∆ (cid:96) + k or, still better, the definition itselfof this distribution shows immediately that β ∗ (∆ (cid:96) + k ) X = Ξ β ( X ) . Thisdistribution Ξ is not, in general, of locally constant dimension but isgenerated by a family of vector fields that is involutive and locally offinite type. In fact, every section of R (cid:96) + k ( S ) determines, by projection,a vector field that is a section of Ξ and, since R (cid:96) + k ( S ) is a Lie equation,the bracket of two sections projects onto the bracket of their images.Moreover, since R (cid:96) + k ( S ) is locally trivial, the pre-sheaf of local sectionsof this fibre bundle is locally of finite type (in fact, locally free) and the local finiteness property of Ξ follows. It is also easy to verify, by usingagain the right translations of Π (cid:96) + k P , that the integral leaves of Ξ areprecisely the projections of the integral leaves of ∆ (cid:96) + k ( cf. [17] and [36],Chap.I, § ). Hence, we thus conclude that β ( R (cid:96) + k ( S ) y ) = B (cid:96) + k ( S ) y is a union of integral leaves of Ξ and this union is discrete: For every z ∈ B (cid:96) + k ( S ) y , there exists an open neighborhood U of z in P suchthat B (cid:96) + k ( S ) y ∩ U reduces to the intersection of U with a unique in-tegral leaf of Ξ . To see this, it suffices to note firstly that the leavesof ∆ (cid:96) + k passing by the points of the same β − fibre of R (cid:96) + k ( S ) y projectall on the same leave of Ξ and secondly, recalling that R (cid:96) + k ( S ) y isa regularly embedded sub-manifold whose connected components areintegral leaves of ∆ (cid:96) + k , we shall take an open neighborhood U , in Π (cid:96) + k ( S ) y , of a point X contained in a β − fibre of R (cid:96) + k ( S ) y in such away that U ∩ R (cid:96) + k ( S ) y just contains the points of a single leaf of ∆ (cid:96) + k .By right translation with the elements of the isotropy R (cid:96) + k ( S ) y , thesame situation reproduces itself, with the translated open set, at everyother point of the β − fibre and consequently U = β ( U ) responds to therequired property. Shrinking, if necessary, the open set U and recall-ing the regularity of the embedding of R (cid:96) + k ( S ) y , we can show furtherthat B (cid:96) + k ( S ) y ∩ U is a slice (in a coordinate system) and, consequently,that B (cid:96) + k ( S ) y is a regularly embedded sub-manifold of P . Finally, asaturation argument of R (cid:96) + k ( S ) y by the action of the total isotropy (Π (cid:96) + k P ) y and the fact that R (cid:96) + k ( S ) y is closed, shows that B (cid:96) + k ( S ) y isclosed in P . We thus see that β : R (cid:96) + k ( S ) y −→ B (cid:96) + k ( S ) y is a surmer-sion (surjective submersion) and, consequently, a (sub-)principal fibrebundle of (Π (cid:96) + k P ) y | B (cid:96) + k ( S ) y that is locally trivial since it admitslocal sections and the proof is therefore complete. We shall neverthe-less continue by showing, further, that the dimension of the eventuallynon-connected sub-manifold B (cid:96) + k ( S ) y is constant on each connectedcomponent of α ( S ) . In fact, since the dimension of R (cid:96) + k ( S ) is constanton each connected component O of α ( S ) , we infer that the dimensionof ∆ (cid:96) + k ( S ) is constant on β − O and, consequently, that of R (cid:96) + k ( S ) y is also constant above O . On the other hand, the tangent space toeach β − fibre of R (cid:96) + k ( S ) y is isomorphic to the Lie algebra R (cid:96) + k ( S ) y of the structural group R (cid:96) + k ( S ) y hence the dimension of this tangentspace is constant. Lastly, since Ξ = T y B (cid:96) + k ( S ) y is isomorphic to thequotient of (∆ (cid:96) + k ) Y , Y ∈ R (cid:96) + k ( S ) y , β ( Y ) = y , modulo the tangentspace to the β − fibre at the point Y , the result follows. To terminate,we provide an alternative proof of the above statements in view ofthe geometrical mechanisms that it will turn apparent and that willbe of relevance in the sequel. For this, we go back to the definition N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 23 of the elements of R (cid:96) + h ( S ) as being the jets j (cid:96) + h ξ ( y ) of local vectorfields ξ whose prolongations pξ are h − th order tangent to the sec-tion S at the point S ( y ) . However, we can see that any Y ∈ R (cid:96) + k ( S ) transforms R (cid:96) + h ( S ) αY into R (cid:96) + h ( S ) βY , h < k , since the k − jet pY with source S ( αY ) and associated, by prolongation, to Y transformsthe k − th order contact element of im S at the point S ( αY ) intothe corresponding contact element at the point S ( βY ) . This property,however, fails when h = k since pY only operates on ( k − − jets ofvector fields on E and the invariance does not subsist any longer, noteven for the projected sub-spaces β ( R (cid:96) + h ( S ) αY ) and β ( R (cid:96) + h ( S ) βY ) .However, if we observe in general, N being an arbitrary manifold, that Π r N operates on J r T N , then it will become apparent ( cf.
Lem-mas 3 and 4) that Y transforms R (cid:96) + k ( S ) αY onto R (cid:96) + k ( S ) βY sincethe jet j k ( pξ )[ S ( αY )] associated to j (cid:96) + k ξ ( αY ) ∈ R (cid:96) + k ( S ) αY belongs to ( J k T E ) S ( αY ) and, consequently, pY transforms the jet j k ( pξ )[ S ( αY )] , k − th order tangent to im S at the point S ( αY ) , into a k − jet ofvector field tangent, up to order k , to im S at the point S ( βY ) , thislater k − jet being precisely the one that corresponds, via prolongation,to the transformed jet Y ( j (cid:96) + k ξ ( αY )) ∈ R (cid:96) + k ( S ) βY . We thus inferthat the fibres of R (cid:96) + k ( S ) at the points αY and βY are isomorphicwhere after the isomorphy of all the fibres along any orbit B (cid:96) + k ( S ) y of R (cid:96) + k ( S ) in P . Since Ξ y = R (cid:96) + k ( S ) y / R (cid:96) + k ( S ) y , we infer that Ξ y has constant dimension along B (cid:96) + k ( S ) y if and only if it is inasmuch for R (cid:96) + k ( S ) y . In particular, this implies the constancy of dimensions for Ξ y on the intersection of B (cid:96) + k ( S ) y with a connected component of αS .The sub-manifold B (cid:96) + k ( S ) y has therefore a constant dimension in eachconnected component of α ( S ) .The method of proof suggests a weakening of the regularity hypothe-ses imposed on R (cid:96) + k ( S ) . Accordingly, it would suffice to assume that R (cid:96) + k ( S ) is locally of finite type i.e. , that it be locally generated by a fi-nite number of sections (differentiable sections of J (cid:96) + k T P taking valuesin R (cid:96) + k ( S ) ). The bracket properties of J (cid:96) + k T P would then imply that R (cid:96) + k ( S ) is a Lie equation, the set of its local sections being closed un-der the bracket. However, in the specific case of the equation R (cid:96) + k ( S ) ,this generalization is only illusory. In fact, the local finiteness of gen-erators would imply lower semi-continuity for dim R (cid:96) + k ( S ) y whereasthe definition of this equation as the kernel of p k Ψ( S ) would imply theupper semi-continuity. In definite, the local finiteness assumption isentirely equivalent to regularity. Micro-differentiable structures and globalization
In this section we look for the hypotheses enabling us to endow thegroupoid R (cid:96) + k ( S ) or eventually its α − connected component with a dif-ferentiable structure compatible with its algebraic structure. To begin,we assume that the vector bundle R (cid:96) + k ( S ) is locally trivial and alreadypossesses all the regularity properties indicated in the last Propositionas well as in its Corollary. We denote by R (cid:96) + k ( S ) the union of allthe integral leaves of ∆ (cid:96) + k issued from the units of Π (cid:96) + k P . A stan-dard connectivity argument shows that R (cid:96) + k ( S ) is a sub-groupoid of R (cid:96) + k ( S ) that we shall call its α − connected component of the unitsspace (assumed to be connected otherwise we argue on each connectedcomponent). Every α − fibre of R (cid:96) + k ( S ) is in fact the connected com-ponent of a unit in the corresponding α − fibre of R (cid:96) + k ( S ) .According to the general definitions ([19], [33]), R (cid:96) + k ( S ) is the sub-groupoid of Π (cid:96) + k P generated by the Lie algebroid R (cid:96) + k ( S ) ⊂ J (cid:96) + k T P .Contrary to what has been said and written in the last century ([34],main theorem), it is well known that Lie’s Second Theorem is inexacteven for transitive Pseudo-groups i.e. , for transitive sub-groupoids ofthe general groupoid Π (cid:96) + k P . In other terms, given a Lie sub-algebroid A of J k T P and denoting by G the (algebraic) sub-groupoid of Π (cid:96) + k P itgenerates ( e.g. , by integrating the α − fibres distribution), it is not al-ways possible to endow this sub-groupoid with a differentiable structure(of sub-manifold) in such a way that its Lie algebroid can be identifiedwith the given one. The main obstruction rests in the non-vanishing holonomy for the integral leaves of ∆ (cid:96) + k , the distribution generatedby the right translations applied to A , these leaves being precisely the α − fibres of the desired sub-groupoid. Nonetheless, in our present sit-uation, this holonomy fortunately vanishes since R (cid:96) + k ( S ) is the kernelof a differential operator (or, more precisely, the kernel of its definingmorphism). We shall of course proceed in the most standard way byfirst endowing an open neighborhood of the units with the differentiablestructure practically "imposed" by the algebroid R (cid:96) + k ( S ) and, there-after, propagate this local differentiable structure to the α − connectedcomponent R (cid:96) + k ( S ) . In order to further propagate this differentiablestructure to the whole of R (cid:96) + k ( S ) , we shall be forced to add an ad-ditional hypothesis. Let us also observe that we are undertaking thispainstaking homework since, to our knowledge, this construction hasnever been fully elucidated before. N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 25
Since ∆ (cid:96) + k is regular and integrable, we take, for each unit e in R (cid:96) + k ( S ) ,a foliating chart ( U α , φ α ) ([14], p.69) in such a way that e ∈ U α and thatthe leaves contained in U α are slices with respect to the coordinate sys-tem φ α , where a slice means a sub-manifold diffeomorphic to an open p − cube in R p , p = dim ∆ (cid:96) + k ( cf. the aforementioned reference). Letus denote by V α the union of all the slices contained in U α and thatcontain units of R (cid:96) + k ( S ) . Then V α is a closed and regularly embeddedsub-manifold of U α . We set U = (cid:83) U α and V = (cid:83) V α . Then V α ∩ V β is an open sub-set of both V α and V β . Consequently, V is a closed andregularly embedded sub-manifold of U and, of course, V ⊂ R (cid:96) + k ( S ) .We next observe that the projection α : V −→ α ( S ) is a surmersionand that the α − fibres contained in V are sub-manifolds of locally con-stant dimension with respect to the variation of y in α ( S ) . Moreover,the germ of this sub-manifold, along the units, is uniquely determined i.e. , does not depend on the initial choice of the foliating charts and,by construction, R (cid:96) + k ( S ) can be identified with the α − vertical tangentbundle of V along the units.Due to its geometrical interest, we shall construct the same germ ofsub-manifold by a procedure relying on the local constancy of the rankof p k Φ( S ) . A unit e ∈ R (cid:96) + k ( S ) being fixed, there exists, due to theconstancy of the rank, a neighborhood U of e in Π (cid:96) + k P such that W = p k Φ( S )( U ) is a regularly embedded sub-manifold of J k E andthat p k Φ( S ) : U −→ W is a surmersion. Furthermore, we can assumethat U = α ( U ) = α ( U ∩ I ) , where I represents the sub-manifold ofunits of Π (cid:96) + k P and, thereafter, the intersection W ∩ im j k S = j k S ( U ) isa regularly embedded sub-manifold of W . The inverse image of j k S ( U ) by the map p k Φ( S ) | U is equal to R (cid:96) + k ( S ) ∩ U and consequently, thisinverse image is a closed and regularly embedded sub-manifold V of U . Since ker T p k Φ( S ) = ∆ (cid:96) + k , we infer that the α − fibres of V areintegral manifolds of maximum dimension of ∆ (cid:96) + k , these α − fibres be-ing precisely the fibres of p k Φ( S ) : U −→ W above the points of j k S ( U ) . By shrinking, if necessary, the open set U , we can be broughtto the case where these fibres are slices and thus infer that any leave of ∆ (cid:96) + k , issued from a point in U ∩ I , intercepts the open set U along aunique slice. Furthermore, the open set U is a foliating chart definedin a neighborhood of the unit e and verifies the conditions stated pre-viously. By taking the union of all these open sets U α correspondingto the various units, we obtain an open neighborhood ˜ U of the unitsof R (cid:96) + k ( S ) in Π (cid:96) + k P such that V = ˜ U ∩ R (cid:96) + k ( S ) = ˜ U ∩ R (cid:96) + k ( S ) isthe union of the slices contained in the open sets U α that contain the units. Besides, V is a neighborhood of the units in R (cid:96) + k ( S ) as well asin R (cid:96) + k ( S ) . The germ, along these units, of the regularly embeddedsub-manifold V is unique and will be called, together with Pradines([32]), the micro-differentiable structure of R (cid:96) + k ( S ) .We now show that this micro-differentiable structure can be extendedto a global structure defined on R (cid:96) + k ( S ) . Indeed, if Y ∈ R (cid:96) + k ( S ) and if e = α ( Y ) , the continuation Theorem ([31], pg.10) enables usto define a differentiable mapping µ : U −→ Π (cid:96) + k P where U isan open neighborhood of the unit e in the sub-manifold I composedby the units, µ ( e’ ) , e’ ∈ U , belongs to the leaf of ∆ (cid:96) + k that con-tains e’ ( α − fibre of R (cid:96) + k ( S ) ) and µ ( e ) = Y . However, if we iden-tify P with the units manifold I , µ becomes a differentiable sec-tion of the bundle α : Π (cid:96) + k P −→ P defined on U with values in R (cid:96) + k ( S ) that assumes the value Y at the point e . Due to the lo-cal rank constancy of p k Φ( S ) , there exists a neighborhood U of Y in Π (cid:96) + k P such that W = p k Φ( S )( U ) is a regularly embedded sub-manifold of J k E and that p k Φ( S ) : U −→ W is a surmersion. Wecan further suppose that U = α ( U ) = α ( U ∩ im µ ) which implies that W ∩ im j k S = j k S ( U ) is a regularly embedded sub-manifold of W . Itthen follow as previously and shrinking, if necessary, the open set U ,that R (cid:96) + k ( S ) ∩ U = R (cid:96) + k ( S ) ∩ U is a closed and regularly embeddedsub-manifold of U , inverse image by the map p k Φ( S ) | U of j k S ( U ) , andthe fibres of the fibration p k Φ( S ) | U coincide, above j k S ( U ) , with the α − fibres of R (cid:96) + k ( S ) ∩ U −→ U . It follows therefore that R (cid:96) + k ( S ) isendowed with the differentiable structure of a regularly embedded sub-manifold of Π (cid:96) + k P compatible with its groupoid structure. Moreover,there exists an open set U in Π (cid:96) + k P such that R (cid:96) + k ( S ) = R (cid:96) + k ( S ) ∩ U or, in other terms, that R (cid:96) + k ( S ) is locally closed. The open set U canbe chosen saturated with respect to the leaves of ∆ (cid:96) + k since R (cid:96) + k ( S ) as well as R (cid:96) + k ( S ) are already saturated. The constructions showclearly that R (cid:96) + k ( S ) identifies with the α − vertical tangent bundle of R (cid:96) + k ( S ) along the units and that, consequently, the sheaf R (cid:96) + k ( S ) isthe Lie algebroid associated to the differentiable sub-groupoid R (cid:96) + k ( S ) ([19],[33]). Proposition 2.
Let S be a structure of species E and finite order (cid:96) verifying the equivalent conditions of the Proposition . The equa-tion R (cid:96) + k ( S ) is then an α − connected and regularly embedded Lie sub-groupoid of Π (cid:96) + k P whose associated Lie algebroid is equal to R (cid:96) + k ( S ) . N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 27
There exists an open set U of Π (cid:96) + k P , saturated by the integral folia-tion of ∆ (cid:96) + k , such that R (cid:96) + k ( S ) is closed in U and, furthermore, thesequence (23) −→ R (cid:96) + k ( S ) −→ U p k Φ( S ) −−−−→ ( J k E, j k S ) is exact. Conversely, when R (cid:96) + k ( S ) admits a structure of Lie sub-groupoid of Π (cid:96) + k P with associated Lie algebroid equal to R (cid:96) + k ( S ) , thenthe equivalent conditions of the Proposition are satisfied and the dif-ferentiable structure of R (cid:96) + k ( S ) is regularly embedded. This proposition states precisely that R (cid:96) + k ( S ) is the non-linear Lieequation generated by the linear Lie equation R (cid:96) + k ( S ) ( cf. [26],[27])when p k Ψ is locally of constant rank. The converse statement of theabove proposition is quite simple to prove and will be omitted. Never-theless, it veils behind the curtains the following rather important fact:We first observe that any local section of R (cid:96) + k ( S ) determines, by righttranslations, a right invariant vector field on Π (cid:96) + k P that is contained in ∆ (cid:96) + k and conversely. Let P denote the pseudo-group of local transfor-mations operating on Π (cid:96) + k P and generated by the flows of the abovementioned right invariant vector field. Then, each element of P leavesinvariant the sub-groupoid R (cid:96) + k ( S ) though it needs not preserve thelarger sub-groupoid R (cid:96) + k ( S ) , this being the main obstacle towards thepossibility of simply extending or prolonging the differentiable struc-ture of the former to the later, as was done in the micro-differentiablesituation. So, let us now take care of R (cid:96) + k ( S ) .In the previous attempt to provide R (cid:96) + k ( S ) with a differentiable struc-ture, there is a key point that still remains unexplored: The open sets U , neighborhood of the unit e , can be chosen in such a way that anyleaf of ∆ (cid:96) + k issued from a point e’ ∈ U ∩ I cuts the open set U alonga unique slice namely, the one containing e’ . We show in fact that U can be chosen in such a way that any leaf of ∆ (cid:96) + k meets U at mostalong a single slice (no holonomy). Since the operations of the groupoid Π (cid:96) + k M are continuous, we can take open neighborhoods V and W of e such that V ⊂ U , W ⊂ U , V · V − ⊂ U and W · V ⊂ U , theoperations being performed on all composable pairs. We can furtherassume that both neighborhoods are the domains of foliating charts,each fibre being a slice. Let us now take a leaf F and assume that itmeets V along two slices S and S . If we take a point X ∈ S then F β ( X ) = F · X − is the leaf of ∆ (cid:96) + k passing by the unit β ( X ) ∈ U ∩ I and consequently ∫ · X − as well as S · X − are included in the sliceof U hence also in that of W and containing the unit β ( X ) . Applyingthe inverse operation · X , we infer that S and S are both containedin a same slice of U and consequently in a same slice of V since thelater is just a foliated chart restriction of U . More generally, we showthat the same property continues to hold at each point of Π (cid:96) + k P , itbeing understood that P is to be replaced by α ( S ) when S is just alocal section. In fact, let X = j (cid:96) + k ϕ ( y ) be an arbitrary point and let usconsider the flow j (cid:96) + k ϕ . By right translations, provided by the flow el-ements, we establish a diffeomorphism τ : β − ( βϕ ) −→ β − ( αϕ ) thatis compatible with ∆ (cid:96) + k and consequently the leaves are transformedin leaves, the foliating charts in foliating charts and the slices in slices.Furthermore, the desired property is verified for the foliating open set τ ( U ) as soon as it is verified for U . In sum, for every X ∈ Π (cid:96) + k P ,there exists a foliating chart U of ∆ (cid:96) + k that is a neighborhood of X andfor which any leaf of ∆ (cid:96) + k meets at most along a single slice. However,these properties translate by saying that the integral foliation of ∆ (cid:96) + k is simple or, in other terms, ([31], pg.19) that there exists a differentiablestructure, necessarily unique, on the quotient Π (cid:96) + k P / ∆ (cid:96) + k of the gen-eral groupoid modulo the integral leaves of ∆ (cid:96) + k and in such a way thatthe quotient map ζ : Π (cid:96) + k M −→ Π (cid:96) + k M / ∆ (cid:96) + k is a surmersion.Moreover, since each leaf of ∆ (cid:96) + k is contained in an α − fibre, there isa canonical projection α of the quotient space onto P that commuteswith α . As previously, we shall also denote by I the identity section y ∈ P (cid:55)−→ j (cid:96) + k Id ( y ) ∈ Π (cid:96) + k P and set I = ζ ◦ I . Then, of course, I is a differentiable section of α , the restriction ζ : im I −→ im I is a diffeomorphism, im I is a regularly embedded sub-manifold ofthe quotient Π (cid:96) + k P / ∆ (cid:96) + k and R (cid:96) + k ( S ) = ζ − ( im I ) . Moreover,since R (cid:96) + k ( S ) is locally closed in a saturated open set U , the sub-manifold im I is locally closed in the open set ζ ( U ) (it should howeverbe observed that Π (cid:96) + k P / ∆ (cid:96) + k needs not be separated). Finally, since R (cid:96) + k ( S ) and R (cid:96) + k ( S ) y are closed and saturated by the leaves and since R (cid:96) + k ( S ) y is a regularly embedded sub-manifold of (Π (cid:96) + k P ) y whose con-nected components are precisely the maximal integral leaves of ∆ (cid:96) + k ,we conclude that ζ ( R (cid:96) + k ( S )) is closed and that ζ ( R (cid:96) + k ( S ) y ) is closedand discrete (each point is isolated) in the fibre (Π (cid:96) + k P / ∆ (cid:96) + k ) y .Let us next assume that R (cid:96) + k ( S ) is endowed with a differentiable struc-ture that makes it become a Lie sub-groupoid of Π (cid:96) + k P and whose as-sociated Lie algebroid is equal to R (cid:96) + k ( S ) . We show that, under theseconditions, the differentiable structure of R (cid:96) + k ( S ) is regularly embed-ded in Π (cid:96) + k P , ζ ( R (cid:96) + k ( S )) becomes a regularly embedded sub-manifold N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 29 of Π (cid:96) + k P / ∆ (cid:96) + k admitting the open subset im I and the restriction α : ζ ( R (cid:96) + k ( S )) −→ α ( S ) is étale .In fact, since R (cid:96) + k ( S ) is a Lie groupoid, the projection α : R (cid:96) + k ( S ) −→ α ( S ) is a surmersion and consequently, for any X ∈ R (cid:96) + k ( S ) , there existsa differentiable local section µ of α taking its values in R (cid:96) + k ( S ) andpassing through X . Furthermore, there exists an open neighborhood V of X in R (cid:96) + k ( S ) such that the fibres of α : V −→ α ( V ) areslices. By the hypothesis, R (cid:96) + k ( S ) is the Lie algebroid of R (cid:96) + k ( S ) hencethe α − vertical tangent bundle V R (cid:96) + k ( S ) is equal to ∆ (cid:96) + k |R (cid:96) + k ( S ) .We infer that the slices of V are integral sub-manifolds of maximumdimension of ∆ (cid:96) + k and, more generally that the α − fibres of R (cid:96) + k ( S ) have, for connected components, the integral leaves of ∆ (cid:96) + k . Let F be the leaf that meets X = µ ( y ) , let us take a second α − section ν of R (cid:96) + k ( S ) such that ν ( y ) ∈ F and let us denote by ˜ V the saturatedset of V , in R (cid:96) + k ( S ) , by the integral leaves of ∆ (cid:96) + k . Furthermore,the continuation Theorem implies that ˜ V is an open subset of R (cid:96) + k ( S ) and, since ν ( y ) ∈ ˜ V , we infer that ν ( z ) and µ ( z ) belong to the sameleaf of ∆ (cid:96) + k as soon as z is sufficiently close to y . However, this impliesthat the two sections ζ ◦ µ and ζ ◦ ν of α coincide in a neighborhoodof y hence enables to define a structure of differentiable sub-manifoldon the image ζ ( R (cid:96) + k ( S )) in such a way that α : ζ ( R (cid:96) + k ( S )) −→ α ( S ) becomes étale . Let us finally show that this sub-manifold is regularlyembedded. To do so, we recall that R (cid:96) + k ( S ) is defined as being thekernel of p k Φ( S ) and that this mapping is locally of constant rank.Therefore, we can find an open neighborhood U of X in (Π (cid:96) + k P ) suchthat W = p k Φ( S )( U ) is a regularly embedded sub-manifold of J k E and p k Φ( S ) : U −→ W is a surmersion. We can further assume that U = α ( µ ) = α ( U ) = α ( U ∩ im µ ) and this implies that W ∩ im j k S = j k S ( U ) is a regularly embedded sub-manifold of W . We infer that R (cid:96) + k ( S ) ∩ U is a closed and regularly embedded sub-manifold of U , inverse imageof j k S ( U ) by the map p k Φ( S ) | U , and whose α − fibres coincide with thefibers of p k Φ( S ) | U above the points of j k S ( U ) . Since, ker p k Φ( S ) =∆ (cid:96) + k , the α − fibres of α : R (cid:96) + k ( S ) ∩ U −→ U are integral sub-manifolds of maximal dimension of ∆ (cid:96) + k and consequently, it is possibleto choose the above open set U as well as the open neighborhood V of X in R (cid:96) + k ( S ) , considered in the beginning, in such a way that V = R (cid:96) + k ( S ) ∩ U and, moreover, that the two differentiable structures, oneinduced by the given structure of R (cid:96) + k ( S ) and the other induced by that of U , be the same and shows consequently that the given structureon R (cid:96) + k ( S ) is regularly embedded. Saturating these open sets by theleaves of ∆ (cid:96) + k , we obtain much in the same way the open and saturatedsub-sets ˜ V = R (cid:96) + k ( S ) ∩ ˜ U and ζ ( R (cid:96) + k ( S ) ∩ U ) = ζ ( R (cid:96) + k ( S ) ∩ ˜ U ) = ζ ( im µ ∩ ˜ U ) = im ( ζ ◦ µ ) ∩ ζ ( ˜ U ) which implies that ζ ( R (cid:96) + k ( S )) is a regularly embedded sub-manifoldof Π (cid:96) + k P / ∆ (cid:96) + k .Conversely, when ζ ( R (cid:96) + k ( S )) admits the structure of a regularly em-bedded sub-manifold, then R (cid:96) + k ( S ) admits the structure of a regu-larly embedded sub-manifold of Π (cid:96) + k P that induces forcefully the reg-ularly embedded structure of R (cid:96) + k ( S ) y . Consequently, the Lie alge-broid associated to the Lie sub-groupoid R (cid:96) + k ( S ) is necessarily equalto R (cid:96) + k ( S ) and this implies again that the structure of ζ ( R (cid:96) + k ( S )) is étale over α ( S ) . It is further clear that im I is an open sub-manifoldof ζ ( R (cid:96) + k ( S )) that however needs not be closed by virtue of the even-tual non-separability of the quotient Π (cid:96) + k P / ∆ (cid:96) + k as well as that of ζ ( R (cid:96) + k ( S )) .We show as well, using the same arguments as above, that if every X ∈R (cid:96) + k ( S ) , the latter without any previously assigned structure, belongsto the image of a differentiable α − section of Π (cid:96) + k P taking its valuesin R (cid:96) + k ( S ) , then R (cid:96) + k ( S ) is a regularly embedded sub-manifold of Π (cid:96) + k P . Such sections are, by the way, usually obtained by composingsections of α : ζ ( R (cid:96) + k ( S )) −→ α ( S ) with sections of ζ . In sum, weproved the following: Proposition 3.
Let S be a structure of species E and finite order (cid:96) satisfying the equivalent requirements of the Proposition . Under theseconditions, the following properties are also equivalent:i) Every element of R (cid:96) + k ( S ) belongs to the image of some differen-tiable section of α : Π (cid:96) + k P −→ P taking its values in R (cid:96) + k ( S ) .ii) The closed subset ζ ( R (cid:96) + k ( S )) ⊂ Π (cid:96) + k P / ∆ (cid:96) + k is a regularlyembedded sub-manifold étale over α ( S ) .iii) R (cid:96) + k ( S ) is a Lie sub-groupoid of Π (cid:96) + k P with associated Lie al-gebroid equal to R (cid:96) + k ( S ) . N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 31 iv) R (cid:96) + k ( S ) is a regularly embedded Lie sub-groupoid of Π (cid:96) + k P . It is desirable, at this point, to examine more carefully the embeddingof R (cid:96) + k ( S ) . With this in mind, let us assume that the structure S sat-isfies the equivalent conditions of the last Proposition whereby R (cid:96) + k ( S ) becomes a regularly embedded Lie sub-groupoid of Π (cid:96) + k P and let usdenote by Π (cid:96) + k P / R (cid:96) + k ( S ) the set of right sided classes of Π (cid:96) + k P mod-ulo R (cid:96) + k ( S ) , that is to say, the quotient modulo the relation: X ∼ Y if and only if X · Y − ∈ R (cid:96) + k ( S ) . These classes are no other than theorbits, by the left action (via the target), of R (cid:96) + k ( S ) on Π (cid:96) + k P andconsequently we see promptly that the right action (via the source) of Π (cid:96) + k P on itself factors to an action (to the right) of Π (cid:96) + k P on the abovequotient Π (cid:96) + k P / R (cid:96) + k ( S ) . We have already seen that each α − fibre of R (cid:96) + k ( S ) is a closed and regularly embedded sub-manifold of an α − fibreof Π (cid:96) + k P and its connected components are the integral leaves of ∆ (cid:96) + k .Let us now take a unit e . Since there exists a saturated open set ˜ U suchthat ˜ U ∩ R (cid:96) + k ( S ) = ˜ U ∩ R (cid:96) + k ( S ) , we infer, in view of the previousresults, that there exists a foliating chart U for ∆ (cid:96) + k , neighborhood of e , such that each α − fibre of R (cid:96) + k ( S ) issued from a point e’ ∈ U ∩ I meets the open set U along a unique slice namely, the slice contain-ing e’ . Let us next observe that the right action of Π (cid:96) + k P permutesthe trajectories (orbits) of R (cid:96) + k ( S ) . Arguments entirely analogous tothose used previously for the integral foliation of ∆ (cid:96) + k will show that,for any X ∈ Π (cid:96) + k P , there exists a foliating chart V of ∆ (cid:96) + k , neighbor-hood of X , such that an arbitrary trajectory of R (cid:96) + k ( S ) will meet theopen set V in at most one slice. We find ourselves within conditionsentirely analogous to those found in the Theorem 8, pg.19 of [31]. Inthis theorem, the differentiable structure of the quotient ( i.e. , the ap-propriate changes of charts) is guaranteed by the transport Theorem([31], pg.10) which however does not apply in the present case in viewof the (eventual) non-connectivity of the α − fibres of R (cid:96) + k ( S ) . Never-theless, the transport can be replaced by the following argument: Let X ∈ R (cid:96) + k ( S ) , e = α ( X ) the corresponding source, µ a differentiablesection of α : R (cid:96) + k ( S ) −→ α ( S ) assuming the value µ ( e ) = X (itis essentially here that intervenes the property ( i ) of the last Proposi-tion) and U , respectively U (cid:48) , the domain of a foliating chart of ∆ (cid:96) + k ,neighborhood of e , respectively X , whose intersection with any orbitof R (cid:96) + k ( S ) reduces at most to a single slice and that verifies moreoverthe condition µ ( U ∩ I ) ⊂ U (cid:48) . Let W be a sub-manifold transverse tothe slices of U such that W ⊃ U ∩ I and let us set V = { Y ∈ W | β ( Y ) ∈ U ∩ I} . Clearly, V remains a transversal sub-manifold and we define the map-ping Σ :
V −→ Π (cid:96) + k P , Σ( Y ) = µ ( βY ) · Y , and we see readily thatthe following properties hold:1) Σ( Y ) is contained in the orbit of Y ,2) Σ | U ∩ I = µ ,3) Σ( V ) ⊂ U (cid:48) , in shrinking if necessary the sub-manifold V .The condition (1) implies that Σ is injective. Furthermore, since theright action of Π (cid:96) + k P on itself is effective and the map µ : U ∩ I −→ µ ( U ∩ I ) is a diffeomorphism of regularly embedded sub-manifolds, we inferthat Σ has injective rank (on the tangent level) and consequently Σ :
V −→ Σ( V ) is a diffeomorphism of sub-manifolds respectingthe orbits (property (1)) and Σ( V is transverse to the slices of U (cid:48) . Wecan therefore extend the conclusions of Palais’ Theorem to the spaceof the orbits of R (cid:96) + k ( S ) since the two charts in the quotient spaceoriginating from U and U (cid:48) do as well originate from the transverse sub-manifolds V and Σ( V ) and therefore are compatible. The quotient set Π (cid:96) + k P / R (cid:96) + k ( S ) admits therefore a (necessarily unique) differentiablemanifold structure for which the quotient map is a surmersion. Wenext remark that Π (cid:96) + k P / ∆ (cid:96) + k = Π (cid:96) + k P / R (cid:96) + k ( S ) and the diagrambelow is commutative, Π (cid:96) + k P / R (cid:96) + k ( S ) ζ ←−− Π (cid:96) + k P ζ (cid:48) −−→ Π (cid:96) + k P / R (cid:96) + k ( S ) ↓ ↓ ↓ P Id ←−− P Id −−→ P the arrow Υ = ζ (cid:48) ◦ ζ − being surjective and étale . Furthermore, R (cid:96) + k ( S ) = ( ζ (cid:48) ) − ( ζ (cid:48) ( I )) (inverse image) and ζ ( R (cid:96) + k ( S )) = Υ − ( ζ (cid:48) ( I )) .The arrows ζ, ζ (cid:48) and Υ are differential co-variants with respect to theright action (by the source) of Π (cid:96) + k P on the three spaces. The quotientmanifold Π (cid:96) + k P / R (cid:96) + k ( S ) can be obtained from Π (cid:96) + k P / R (cid:96) + k ( S ) byidentifying the points on each orbit that are deducible one from the N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 33 other by the discrete action of R (cid:96) + k ( S ) and this identification is glob-ally compatible when the equivalent properties of the above Propositionare verified. We next observe that the co-variance of p k Φ( S ) enables thefactorisation of this morphism, modulo the action of R (cid:96) + k ( S ) , and con-sequently the diagram that follows is commutative, the factored differ-ential co-variant p (cid:48) k Φ( S ) becoming an injective immersion ( inmersion?Kkkk ). Π (cid:96) + k P p k Φ( S ) −−−−→ J k E (24) ζ (cid:48) ↓ (cid:37) p (cid:48) k Φ( S )Π (cid:96) + k P / R (cid:96) + k ( S ) We infer that im p k Φ( S ) is a sub-manifold (not always regularly embed-ded) of J k E canonically isomorphic to the quotient Π (cid:96) + k P / R (cid:96) + k ( S ) ,the map α : im p k Φ( S ) −→ α ( S ) is a fibration and p k Φ( S ) : Π (cid:96) + k P −→ im p k Φ( S ) is a surmersive morphism of fibrations having for basis α ( S ) ( P beingreplaced by α ( S ) when S is not global). The covariance of p k Φ( S ) finally shows that ( im p k Φ( S ) , α, α ( S ) , p k ) is a prolongation space oforder (cid:96) + k and that the map p (cid:48) k Φ( S ) : (Π (cid:96) + k P / R (cid:96) + k ( S ) , α, α ( S ) , p Sk ) −→ ( im p k Φ( S ) , α, α ( S ) , p k ) is an isomorphism of prolongation spaces where the first term is giventhe quotient prolongation space structure, modulo the right action of R (cid:96) + k ( S ) , of the canonical structure of Π (cid:96) + k P resulting from the stan-dard prolongation operation by the source ([20], §
16, part (b)).
Theorem 1.
Let S be a structure of species E and of finite order (cid:96) suchthat p k Ψ( S ) is locally of constant rank. Then the following conditionsare equivalent:i) R (cid:96) + k ( S ) is a Lie sub-groupoid of Π (cid:96) + k P whose associated Liealgebroid is equal to R (cid:96) + k ( S ) .ii) R (cid:96) + k ( S ) is a regularly embedded (and closed) Lie sub-groupoid of Π (cid:96) + k P .iii) There exists a differentiable structure on Π (cid:96) + k P / R (cid:96) + k ( S ) suchthat the quotient map is a submersion. iv) The image of p k Φ( S ) admits a sub-manifold structure of J k E such that the map p k Φ( S ) : Π (cid:96) + k P −→ im p k Φ( S ) is a submersion.v) Every element of R (cid:96) + k ( S ) belongs to the image of a local differ-entiable section of α : Π (cid:96) + k P −→ P taking values in R (cid:96) + k ( S ) .These equivalent conditions being verified, the quotient differential co-variant p (cid:48) k Φ( S ) : (Π (cid:96) + k P / R (cid:96) + k ( S ) , α, α ( S ) , p Sk ) −→ ( im p k Φ( S ) , α, α ( S ) , p k ) is an isomorphism of prolongation spaces. Corollary 2.
Let S be a structure of species E and of finite order (cid:96) such that p k Ψ( S ) is locally of constant rank and let us assume furtherthat the equation R (cid:96) + k ( S ) is transitive ( β ( R (cid:96) + k ( S )) = T P ) . Underthese conditions, the equivalent properties of the previous Theorem arealways satisfied namely, R (cid:96) + k ( S ) is a closed and regularly embeddedLie sub-groupoid of Π (cid:96) + k P . Moreover, R (cid:96) + k ( S ) as well as R (cid:96) + k ( S ) are locally trivial Lie sub-groupoids and R (cid:96) + k ( S ) is closed in Π (cid:96) + k P . Proof . The transitivity of R (cid:96) + k ( S ) implies that the restriction of β to each α − fibre of R (cid:96) + k ( S ) is a submersion (that will be surjectivewhenever R (cid:96) + k ( S ) becomes transitive). Let X ∈ R (cid:96) + k ( S ) , y = α ( X ) ,and let us take a section σ of β : R (cid:96) + k ( S ) x −→ P defined on an openneighborhood U of y such that σ ( y ) = e (the unit associated to y ). Themap τ : U −→ Π (cid:96) + k P , τ ( y ) = X · σ ( y ) − , is an α − section taking itsvalues in R (cid:96) + k ( S ) and such that τ ( y ) = X , thus retrieving the property( v ) of the last Theorem. The submersivity of β on each α − fibre of R (cid:96) + k ( S ) implies the submersivity of α × β : R (cid:96) + k ( S ) −→ α ( S ) × α ( S ) whereupon the possibility ([19]) in defining local trivialisationsof R (cid:96) + k ( S ) and R (cid:96) + k ( S ) with the help of local sections of α × β .We also remark, en passant , that each domain of a local trivialisationadmits a regularly embedded differentiable structure inherited from theisotropy, at a given point, of R (cid:96) + k ( S ) (resp. R (cid:96) + k ( S ) ) , this isotropybeing a closed Lie sub-group hence regularly embedded (resp. regularlyembedded hence also closed). The family of all such trivialisations iscompatible and defines thereafter the regularly embedded structure of R (cid:96) + k ( S ) inasmuch as that of R (cid:96) + k ( S ) . We finally note that, in thetransitive case envisaged, the topological nature of R (cid:96) + k ( S ) is entirelydetermined by the topological nature of its isotropy group at a point.Since this group is closed in (Π (cid:96) + k P ) y , we derive that R (cid:96) + k ( S ) is N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 35 closed in Π (cid:96) + k P . Observe, however, that the isotropy of R (cid:96) + k ( S ) isnot necessarily the connected component, of the unit, in the isotropyof R (cid:96) + k ( S ) . Corollary 3.
Let S be a structure of species E and of order (cid:96) such that p k Ψ( S ) is locally of constant rank. Under these conditions,a) R (cid:96) + k + h ( S ) is the standard h − th prolongation of R (cid:96) + k ( S ) i.e., R (cid:96) + k + h ( S ) = Π (cid:96) + k + h P ∩ J h R (cid:96) + k ( S ) . b) If, moreover, S verifies the equivalent conditions of the Theorem,then R (cid:96) + k + h ( S ) is the standard h − th prolongation of R (cid:96) + k ( S ) . The proof of this corollary relies on [22] and on the following Lemma:
Lemma 5.
Let E (cid:48) −→ E −→ ( E (cid:48)(cid:48) , σ ) be an exact sequence of fibrationsover the base space P (exact in the set theoretical sense and also in thevertical tangential sense). Then, for any integer k, the sequence ofprolonged fibrations J k E (cid:48) −→ J k E −→ ( J k E (cid:48)(cid:48) , j k σ ) is also exact. It is then achieved by an inductive argument on the integer k using, ateach stage, the affine structure of the kernels as well as the exactness ofthe sequence of linear symbols that is a consequence of the tangentialexactness of the initially given sequence.In order to prove the part ( b ) of the corollary, we simply use the ex-actness of the sequence R (cid:96) + k ( S ) −→ Π (cid:96) + k P p k Φ( S ) −−−−→ ( J k E, j k S ) whose tangential exactness follows from the local constancy of the rankof p k Φ( S ) and thereafter observe that the diagram below is commuta-tive and exact: −→ J h R (cid:96) + k ( S ) −→ J h Π (cid:96) + k P J h p k Φ( S ) −−−−−→ ( J h J k E, j h j k S ) (cid:37) ι ↑ ι ↑ p h p k Φ( S ) ι ↑ (cid:30) −→ R (cid:96) + h + k ( S ) −→ Π (cid:96) + h + k P p h + k Φ( S ) −−−−−→ ( J h + k E, j h + k S ) ↑ ↑ As for the part ( a ), we simply replace, according to the Proposition 2,the previous exact sequence by R (cid:96) + k ( S ) −→ U p k Φ( S ) −−−−→ ( J k E, j k S ) We now observe that it is essential to use hypotheses guaranteeing theappropriate differentiable structures for R (cid:96) + k ( S ) and R (cid:96) + k ( S ) in thelack of which the above Corollary becomes inexact, not subsisting butthe inclusion Π (cid:96) + h + k P ∩ J h R (cid:96) + k ( S ) ⊂ R (cid:96) + h + k ( S ) . The above proof being rather esotheric , we shall transcribe it in lo-cal coordinates for the usage of the non-initiated. However, this naivetranscription will be useful later. We recall that both Lie and Car-tan frequently indulged into incredible calculations since they believed,presumably, that this was the first step in understanding Heaven righthere from earth. Nowadays, calculations are for many a boring activitythough, for others, become indispensable. Just imagine a cosmologisttrying to figure out whether Einstein’s constant c is the same here inour vicinity as in, for example, Andromeda , 60 million light years away,or in the
Whirlpool Galaxy ? The same doubts arise inasmuch for π andon how do Pythagorean circles behave in the Magellania Cloud or onhow does e behave, does it also change + for × , in Antennae ? Evenat a much closer range, just beyond the neutral zone, we might askwhether the Klingon’s uncertainty (undecidability) (cid:126) is as high as forthe humans or whether they are more self-confident? ([28])In order to do so, let us return to the notations in the proof of the lastProposition and take a point X ∈ R (cid:96) + k ( S ) as well as a differentiable α − section µ : U −→ Π (cid:96) + k P taking its values in R (cid:96) + k ( S ) and such that µ ( y ) = X . By the local constancy of the rank of p k Φ( S ) , there exists anopen neighborhood U of X in Π (cid:96) + k + h P such that W = p k Φ( S )( U ) is aregularly embedded sub-manifold of J k E and that p k Φ( S ) : U −→ W qualification donnée, dans les écoles des anciens philosophes, à leure doctrinesecrète, réservée aux seuls initiés N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 37
Figure 1.
The Universeis a surmersion. We can assume, as previously, that U = α ( U ) = α ( U ∩ im µ ) which implies that W ∩ im j k S = j k S ( U ) is a regularlyembedded sub-manifold of W . Shrinking, if necessary, the open set U , let us take a finite family ( f i ) of independent functions whose zerosdefine the sub-manifold j k S ( U ) in W . The local constancy of therank of p k Φ( S ) combined with the key property W ∩ im j k S = j k S ( U ) forces the composite functions F i = f i ◦ p k Φ( S ) to be also independent,their zeros defining R (cid:96) + k ( S ) ∩ U . We shall now complete the functions ( F i ) to a local coordinates system by adding some functions ( G j ) . Thecondition Y = j (cid:96) + h + k ϕ ( y ) ∈ R (cid:96) + h + k ( S ) means that p h + k Φ( S )( Y ) = j h + k S ( y ) or, equivalently, since p h + k = ( p k ) h , that j h ( p k Φ( S )( j (cid:96) + k ϕ ))( y ) = j h ( j k S )( y ) . Translated into coordinates, this condition reads j h ( F i ◦ j (cid:96) + k ϕ )( y ) = 0 .However, still in coordinates, if we replace j (cid:96) + k ϕ by the local sec-tion σ of α : R (cid:96) + k ( S ) −→ P whose components along the coor-dinates F i are null and, along the G j , are equal to those of j (cid:96) + k ϕ ,we shall obtain the equality j h ( j (cid:96) + k ϕ )( y ) = j h σ ( y ) and consequently Y ∈ Π (cid:96) + h + k P ∩ J h R (cid:96) + k ( S ) . The above argument is in fact a transver-sality argument of p k Φ( S ) with im j k S in a slightly more general con-text since transversality of the two, in the usual context, does not hold.What holds in fact is the following: There exists, in an open neighbor-hood W of each point y belonging to im j k S , a local foliation forwhich one of its leaves is an open subset of im j k S (for example, thefoliation f i = c i of the former open set W ) and such that, if we denoteby W (cid:48) the quotient of W modulo the leaves and by ρ : W −→ W (cid:48) theprojection, the composed map ρ ◦ p k Φ( S ) will be of constant rank on asufficiently small open neighborhood V of y . This argument proves the claim in part ( b ) of the Corollary. As for the part ( a ) it will suffice torepeat the argument placing us above the open set U .Let us now reassume the general case where no regularity hypothe-sis, on p k Ψ( S ) , is assumed. We already remarked that the isotropies ( R (cid:96) + k ( S )) y and ( R (cid:96) + k S ) y only depend upon the jet j k S ( y ) and that, inparticular, the isotropy of order (cid:96) only depends on the point S ( y ) . Itthen follows that the symbol ( g (cid:96) S ) y of R (cid:96) ( S ) at the point y also dependsonly on S ( y ) . Observing that R (cid:96) + k ( S ) = ker p k Ψ( S ) , a simple calcu-lation ([20], Lemma 23.2, [23], Proposition 4.3, [35], Proposition 9.3)will show that the symbol ( g (cid:96) + k S ) y of R (cid:96) + k ( S ) is the k − th algebraicprolongation ( espace déduit ) of the symbol ( g (cid:96) S ) y and consequently, Proposition 4.
The symbol of R (cid:96) + k ( S ) at the point y only dependsupon S ( y ) ∈ E and this result is independent of any regularity conditionrequirement on the morphism p k Ψ( S ) . Moreover, the symbol ( g (cid:96) + k S ) y is the k − th algebraic prolongation of ( g (cid:96) S ) y . We now examine the non-linear situation. Let X ∈ R (cid:96) + k − ( S ) , considerthe canonical projection ρ : Π (cid:96) + k P −→ Π (cid:96) + k − P and define the non-linear symbol ( g (cid:96) + k S ) X = { Y ∈ R (cid:96) + k ( S ) | ρ ( Y ) = X } of R (cid:96) + k ( S ) above X . Let us show and this without any regularity hy-potheses on p k Φ( S ) that g (cid:96) + k S ) X is an affine sub-space of the totalsymbol above X namely, the space { Y ∈ Π (cid:96) + k P | ρ ( Y ) = X } . We firstobserve that, for each pair ( y, z ) ∈ P × P , the set R (cid:96) + k ( S ) ( y,z ) = { X ∈ R (cid:96) + k ( S ) | α ( X ) = y, β ( X ) = z } is a simply transitive homogeneous space of the group R (cid:96) + k ( S ) y bythe right action and also of the group R (cid:96) + k ( S ) z by the left action. Itthen follows that R (cid:96) + k ( S ) ( y,z ) is a closed and regularly embedded sub-manifold of Π (cid:96) + k P ( y,z ) canonically isomorphic to the left or to the rightisotropy. For any X ∈ R (cid:96) + k − ( S ) , with y = α ( X ) and z = β ( X ) ,the same argument shows that the symbol ( g (cid:96) + k S ) X is a closed andregularly embedded sub-manifold of the total symbol above X since itis a simply transitive homogeneous space of the closed Lie group ( g (cid:96) + k S ) y = ker [ R (cid:96) + k ( S ) y −→ R (cid:96) + k − ( S ) y ] N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 39 by the right action and also of the corresponding group ( g (cid:96) + k S ) z by theleft action. Note that ( g (cid:96) + k S ) y is simply the symbol of R (cid:96) + k ( S ) abovethe unit of R (cid:96) + k − ( S ) that identifies with y . Let us finally show that ( g (cid:96) + k S ) X is a connected sub-manifold, in fact an affine sub-space ofthe total symbol. For this, let Z be the projection of X in R (cid:96) ( S ) andobserve that every Y ∈ ( g (cid:96) + k S ) X projects onto Z by the projection ρ (cid:96) . We now take ( at last ) local coordinate systems ( y i ) in an openneighborhood U of y and ( z λ ) in an open neighborhood V of z andconsider the corresponding "jet" coordinate system ( y i , z λ , z λα ) | α |≤ r onthe open set ( α × β ) − r ( U × V ) of Π r P . Inasmuch, we also take anadapted local coordinate system ( y i , w µ ) in an open neighborhood W of the point Φ( S )( Z ) = S ( y ) in E ( y i = y i ◦ π ) and there existsof course an open neighborhood U (cid:96) of Z in ( α × β ) − (cid:96) ( U × V ) suchthat Φ( S )( U (cid:96) ) ⊂ W . We write { Φ i , Φ µ } the components, along thecoordinates ( y i , w µ ) , of the restriction Φ( S ) : U (cid:96) −→ W . Since Φ( S ) is a morphism over the Identity, we infer that Φ i ( y i , z λ , z λα ) = y i .We denote by U (cid:96) + k the inverse image, by ρ : Π (cid:96) + k P −→ P i (cid:96) P , ofthe open set U (cid:96) and observe that the former contains ( g (cid:96) + k S ) X andis endowed with the restrictions of the coordinates ( y i , z λ , z λα ) | α |≤ (cid:96) + k .Finally, denote by ρ − ( W ) = W k the open set, of J k E , inverse image of W and equipped with the natural coordinates ( y i , w µ , w µα ) | α |≤ k derivedfrom ( y i , w µ ) . Then p k Φ( S )( U (cid:96) + k ) ⊂ W k and the components of therestriction p k Φ( S ) : U (cid:96) + k −→ W k , with respect to the coordinates ( y i , w µ , w µα ) | α |≤ k , are precisely the functions { Φ i , Φ µ , ∂ α Φ µ } | α |≤ k where ∂ α is the total derivative of order | α | with respect to the variables ( y i ) (iterated total derivatives in jet spaces). In particular, the symbol ( g (cid:96) + k S ) X is defined, in the affine space of the total symbol over X , bythe equations: ∂ α Φ µ ( Y ) = w µα ( j k S ( x )) = ( ∂ α S µ /∂y α )( y ) , | α | = k . Since ∂ α Φ µ ( Y ) = (cid:88) | β | = (cid:96) ( ∂ Φ µ /∂z λβ )( Z ) · z λα + β + F α ( X ) , where, on the right hand side, we only detail the highest order terms,the symbol ( g (cid:96) + k S ) X being thereafter determined by the following lin-ear equations with constant coefficients in the variables z λγ , | γ | = (cid:96) + k , (cid:88) | β | = (cid:96) ( ∂ Φ µ /∂z λβ )( Z ) · z λα + β = ( ∂ α S µ /∂y α )( y ) − F α ( X ) , defining, as pretended, a linear affine sub-space in the space of the totalsymbol.Let us now glimpse at the intrinsical aspects. With the help of thecanonical identification, we can see that the abelian Lie algebra ( g (cid:96) + k S ) z ⊂ R (cid:96) + k ( S ) z is not only the Lie algebra of the group ( g (cid:96) + k S ) z but hasmuch more impact. Recalling the results of [20], §
19, each element v ∈ g (cid:96) + k S ) z determines a vector field on the total symbol space thatgenerates a global 1-parameter group ( ϕ t ) t with the property that ϕ ( Y ) = Y + v is precisely the affine operation by the vector v . The or-bits of this action are all the linear affine sub-spaces of the total symbolwhose direction is given by ( g (cid:96) + k S ) z and finally, since the sub-spacesgenerated, at each point, by the above vector fields are necessarily con-tained in ∆ (cid:96) + k , the Proposition 1 will imply that an orbit, by the above(infinitesimal) affine action of ( g (cid:96) + k S ) z , that contains an element of ( g (cid:96) + k S ) X is entirely contained in ( g (cid:96) + k S ) X . A dimensional argumentwill also show that dim ( g (cid:96) + k S ) X is equal to the dimension of the or-bits and the previous tinkering ( bricolage ) with local coordinates onlyserves to prove that ( g (cid:96) + k S ) X is also connected hence equal to an entireorbit. Furthermore, if we consider as symbol of a non-linear equationthe family of all tangent spaces to the non-linear symbol ( g (cid:96) + k S ) X , thetangent symbol , we perceive that this family of tangent symbols is noth-ing else, by the canonical identification, than ( g (cid:96) + k S ) z . It then followsthat the tangent symbol of R (cid:96) + k ( S ) above the point X is the k − th algebraic prolongation of the tangent symbol of R (cid:96) ( S ) at the point Z ,hence only depends on S ( z ) . Otherwise, this last result can equally beobtained with the help of the Lemma 23.2 in [20] or the Proposition4.3 in [23] or still the Proposition 9.3 in [35]. Let us finally observethat the groupoid structure of R (cid:96) + k ( S ) enables us to further explicitthe affine structure of the non-linear symbol. In fact, the argument incoordinates shows that the group ( g (cid:96) + k S ) z is connected, non-compactand actually homeomorphic to a numerical space. Being abelian, itcanonically identifies with its Lie algebra ( g (cid:96) + k S ) z and the affine ac-tion of the latter on the total symbol above X is nothing else but theleft action by the abelian group ( g (cid:96) + k S ) z . The symbol ( g (cid:96) + k S ) X is justone of the orbits of this action, the restricted action becoming simplytransitive (without fixed points). Similarly, the right action of ( g (cid:96) + k S ) y on ( g (cid:96) + k S ) X is an affine space structure that coincides with the previ-ous one as soon as we identify ( g (cid:96) + k S ) y with ( g (cid:96) + k S ) z by means of aconjugation via an element of ( g (cid:96) + k S ) X . N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 41
Proposition 5.
Without any regularity hypotheses on p k Φ( S ) , thesymbol of R (cid:96) + k ( S ) above any point X ∈ R (cid:96) + k − ( S ) is an affine sub-space of the total symbol and the corresponding affine structure can beidentified with the left action by the symbol ( g (cid:96) + k S ) z of the isotropy atthe point z = β ( X ) . The tangent symbol above X is isomorphic to ( g (cid:96) + k S ) z and consequently only depends upon the point S ( z ) ∈ E . Inparticular, the non-linear symbol and the tangent symbol above a unit e ∈ R (cid:96) + k − ( S ) i.e., the symbol of the isotropy group of order (cid:96) + k at the point y = α ( e ) = β ( e ) as well as its Lie algebra ( g (cid:96) + k S ) y onlydepend upon S ( y ) . The general problem
Let ( E, π, P, p ) be a finite prolongation space of order (cid:96) and let usnow assume that E hence consequently P are paracompact spaces.Let us denote by Γ = Γ( P ) the general pseudo-group of all the lo-cal diffeomorphisms of P and by L = L ( P ) the pseudo-algebra of allthe local vector fields (infinitesimal automorphisms). By prolongationof Γ (resp. L ) to J k E , we obtain the pseudo-group Γ (cid:96) + k resp., thepseudo-algebra (pre-sheaf of Lie algebras) L (cid:96) + k . These are obtainedby localisation of p k Γ resp., p k L which means that we consider the setof all local finite or infinitesimal transformations of J k E that coincidelocally with the prolonged transformations (and where p k = p k ◦ π ).Although Γ and L are "Lie" at any order, this might fail to be truewith the prolonged objects, the regularity of these being closely relatedto the geometry of the prolongation space E . Nevertheless, we can stillobtain much information concerning the formal equivalence problem aswell as on other matters involving structures of species E by examiningclosely the trajectories (orbits) of these prolonged pseudo-groups andpseudo-algebras. In fact, the "Lemma" is as follows: Two k-jets of structures of species E are equivalent (or two germs ofstructures of species E are equivalent up to order k) when the two jetsfind themselves on the same trajectory of Γ (cid:96) + k .The prolongation space p k : J k E −→ P being of order (cid:96) + k , weknow that any local diffeomorphism ϕ of P prolongs to a local dif-feomorphism p k ϕ defined by p k ϕ ( X ) = j k ( pϕ ) · X . We thus obtaina left or right action of the groupoid Π (cid:96) + k P on J k E though, for thetime being, we only consider the left action. If Z · X = Z (cid:48) · X ,then of course Z − Z (cid:48) ∈ Π (cid:96) + k P X , the isotropy group of Π (cid:96) + k P at thepoint X , that we shall denote by H (cid:96) + k ( X ) or simply H ( X ) . The ac-tion being differentiable, each isotropy group is a closed Lie subgroup of Π (cid:96) + k P y , y = α ( X ) and the isotropies at two distinct points ofthe same orbit Ω( X ) = Π (cid:96) + k P · X are conjugate subgroups. Further-more, the quotient space (Π (cid:96) + k P ) y / H (cid:96) + k ( X ) of the classes, to the left,of (Π (cid:96) + k P ) y that are orbits under the right action (by the source) of H (cid:96) + k ( X ) , is a differentiable fibre bundle in homogeneous spaces via theleft action (by the target) of Π (cid:96) + k P ) z , z = β ( X ) , on the fibre above z . The isotropy of this left action at the point Z ∈ H ( X ) is equal to H ( Z · X ) , it is obtained by the conjugation H ( Z · X ) = ZH ( X ) Z − and the diagram below is commutative: (Π (cid:96) + k P ) y q −→ Ω( X ) ↓ (cid:37) µ (Π (cid:96) + k P ) y / H (cid:96) + k ( X ) the arrow µ being bijective onto Ω( X ) and differentiable as a mappinginto J k E . Let us now transport on Ω( X ) and by means of µ the dif-ferentiable structure coming from the quotient and show that Ω( X ) becomes a sub-manifold of J k E though not necessarily regularly em-bedded. To do so, it will suffice to show that µ , as a mapping into J k E , is an immersion (maximum injective rank) and this leads us toexamine the infinitesimal prolongation.The prolonged infinitesimal pseudo-algebra L (cid:96) + k induces, at each point,a subspace of the tangent space to J k E and consequently a distribution(field of contact elements) ∆ k on the manifold J k E that is generatedby a family of vector fields stable under the bracket. Though involu-tive, this distribution can admit singularities. Since the infinitesimalprolongation operator p k is of order (cid:96) + k , each subspace (∆ k ) X is en-tirely determined by ( J (cid:96) + k T P ) y , y = α ( X ) and, more precisely, thefollowing sequence is exact: J (cid:96) + k T P × P J k E λ k −→ ∆ k −→ Since the sheaf of germs of local sections of J (cid:96) + k T P is free and of finiterank, the image sheaf, that is closed for the bracket and generates ∆ k ,is also of finite type hence ([17],[36]) every X ∈ J k E is contained ina maximal integral sub-manifold ω ( X ) and verifies T Y ω ( X ) = (∆ k ) Y for all Y ∈ ω ( X ) , though the ensemble of these integral sub-manifoldsdoes not form necessarily a regular foliation since their dimensions canvary. The space J k E admits therefore a partition by integral leaves,with eventual singularities, of ∆ k . Moreover, the leaf ω ( X ) is the set N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 43 of points of J k E that can be joined from X by piece-wise differentiableintegral curves of ∆ k . Since ∆ k is generated by L (cid:96) + k , the leaf ω ( X ) isalso the trajectory of L (cid:96) + k passing by X hence, the set of all points of J k E that we can join to X (or, for that matter, from X ) by piece-wisedifferentiable curves where each differentiable arc is the trajectory of avector field belonging to L (cid:96) + k . We infer that ω ( X ) ⊂ Ω( X ) and, moregenerally, that Ω( X ) is a union of integral leaves of ∆ k . Furthermore,since J (cid:96) + k T P is the Lie algebroid of the Lie groupoid Π (cid:96) + k P and sincea)the prolongation, by the target, of L to Π (cid:96) + k P is infinitesimallytransitive on each α − fibre, the trajectories of the prolonged algebroid L (cid:96) + k being the connected components of the α − fibres - maximal inte-gral sub-manifolds of the trivial distribution V Π (cid:96) + k P - as well asb) the infinitesimal action p k being derived from the finite action p k ,we infer (always under the canonical identification) thatc) the tangent map to (Π (cid:96) + k P ) y −→ Ω( X ) ⊂ J k E at the point Z is equal to λ k : ( J (cid:96) + k T P ) β ( Z ) −→ T Z · X J k E and consequently its rankis equal to dim (∆ k ) Z · X ,d) the Lie algebra h ( X ) of H ( X ) is equal to the kernel of the map λ k : ( J (cid:96) + k T P ) α ( X ) −→ T X J k E , ande) since ZH ( X ) = H ( Z · X ) Z , the kernel of the tangent map to (Π (cid:96) + k P ) y −→ (Π (cid:96) + k P ) y / H (cid:96) + k ( X ) at the point Z is equal to h ( Z · X ) = ker ( λ k ) α ( Z · X )= β ( Z ) ,we conclude that ( µ ∗ ) ZH ( X ) : ( J (cid:96) + k T P ) β ( Z ) / h ( Z · X ) −→ (∆ k ) Z · X is an isomorphism hence Ω( X ) is a sub-manifold of J k E for which theintegral leaves of ∆ k are open sets. Since these leaves are the trajecto-ries of L (cid:96) + k , they are disjoint and constitute the connected componentsof Ω( X ) . Finally, since the connected components of Π (cid:96) + k P y are thetrajectories of the standard prolongation of L by the target, we seethat the image of each connected component of Π (cid:96) + k P y by the map q is an integral leaf of ∆ k contained in Ω( X ) and therefore the inverseimage of a leaf is a union of connected components. In particular, the image of the connected component of the unit at the point y is equalto ω ( X ) . Theorem 2.
Each orbit Ω( X ) of Γ (cid:96) + k is a differentiable sub-manifoldof J k E canonically isomorphic to Π (cid:96) + k P y / H (cid:96) + k ( X ) and invariant un-der L (cid:96) + k , the infinitesimal action being transitive. The quadruple (Ω( X ) , α, P, p k ) is a finite prolongation space of order (cid:96) + k and thegroupoid Π (cid:96) + k P as well as the sheaf (pseudo-algebra) J (cid:96) + k T P operateon it. The restrictions of Γ (cid:96) + k and L (cid:96) + k to Ω( X ) are finite and in-finitesimal pseudo-groups and pseudo-algebras of arbitrary order. Forall k ≥ h , the canonical projection ρ h,k transforms every k − th orderorbit onto an h − th order orbit and thus defines a prolongation spacesmorphism. The distribution ∆ k on J k E induced by L (cid:96) + k is involutiveand locally of finite type, its maximal integral manifolds are the orbits of L (cid:96) + k and each orbit of the finite action has for its connected componentsthe orbits of the infinitesimal action. The quadruple ( ω ( X ) , α, P, p k ) isan infinitesimal prolongation space of order (cid:96) + k whenever P is con-nected. The standard prolongation by the target ([20], §
16, item (a)) deter-mines a canonical finite prolongation structure (Π k P, β, P, p bk ) of finiteorder k for which the α − fibres are the trajectories . For each y ∈ P , theprolongation sub-space ((Π k P ) y , β, P, p bk ) is transitive and the equiva-lence relation defined by any closed Lie sub-group H ⊂ (Π k P ) y iscompatible with the prolongation operations. Consequently, the quo-tient quadruple ((Π k P ) y / H, β, P, p bk ) is a finite prolongation space oforder k . Corollary 4.
The canonical isomorphism of the preceding theorem isan isomorphism of prolongation spaces ((Π (cid:96) + k P ) y / H ( X ) , β, P, p b(cid:96) + k ) µ −−→ (Ω( X ) , α, P, p k ) . Since ∆ k admits in general singularities, the space of orbits by the finiteor infinitesimal actions is most often rather complicated. It can reduceto a finite number or to a discrete family of orbits (for the quotienttopology), it can present itself as a regular foliation (continuous familyof orbits) and, most often, the two options can appear simultaneously. p sk - prolongement par la source , p bk - prolongement par le but . N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 45
The discrete orbits correspond geometrically to the existence of mod-els (in coordinates) for the germs of structures or for their k − jets.Quite to the contrary, the continuous families of orbits apparently turnnonexistent the presence of models these being replaced by local de-formations of non-equivalent structures, since the nature in itself of amodel highlights and emphasizes the notion of rigidity (not to be con-founded with the deformation of structures locally equivalent to a givenmodel). When the orbits are discrete, the formal and local equivalenceproblems will have to be examined by methods specific to each caseand using all the available techniques as well as the invariants. Thisis the case especially for the "modeled structures" (for example, mod-eled on a Lie pseudo-group) where we shall first try to establish theformal equivalence with the model and thereafter the local equivalenceleading most often to an integrability problem. When the orbits aredistributed along continuous families, it seems advantageous to appealto the differential invariants of the action of Γ on the space E . Thesehowever are only speculations and the sole positive statement is thefollowing: Two infinite jets of structures of species E are formally equivalent ifand only if their k − jets, for any k, belong to the same k − th orderorbit. The restricted problem
Often, mainly in physics and other domains, it is important to knowthe equivalence not only with respect to an arbitrary transformationbut also one respecting certain additional properties ( e.g. , conservationlaws) and this conveys us to what we call the restricted equivalenceproblem with respect to the transformations of a given pseudo-groupor pseudo-algebra.Let Γ be a pseudo-group of local transformations operating on the man-ifold P and L the corresponding infinitesimal pseudo-algebra (some-times called infinitesimal pseudo-group) i.e. , the sub-presheaf of Γ( T P ) (sections) obtained by localizing as well as pasting together all the lo-cal vector fields of P of the form ξ = ddt ϕ t | t =0 , where ( ϕ t ) t is a localone parameter family of element of Γ . We shall say that Γ is a Liepseudo-group of order k if, for any k ≥ k , the following propertieshold:a) J k Γ is a closed Lie sub-groupoid of Π k P and the projection ρ : J k + h Γ −→ J k Γ is a surmersion.b) J k +1 Γ , considered as a differential equation on the fibration α : Π k +1 P −→ P , is the standard prolongation of J k Γ .c) J k Γ is infinitesimally complete i.e. , the associated linear Lie equa-tion R k = V J k Γ | P ( P being identified with the units of J k Γ and V = α − vertical ) is equal to J k L .d) Γ is complete of order k which means that ϕ ∈ Γ if and only if j k ϕ is a section (solution) of J k Γ . Remark:
When (a) is verified, we can easily prove that J k +1 Γ is con-tained in the prolongation (as a differential equation) of J k Γ and conse-quently the property (b) will follow eventually at a higher order k + h namely, when the δ − cohomology of the symbols of the linear equations R k become − acyclic. This results locally, in an open neighborhood U of a point in J m +1 Γ , m = k + h , in virtue of the prolongation theo-rem of Cartan-Kuranishi ([23], Theorem 10.1). The invariance of theprolongation p ( J m Γ) by the left action of J m +1 Γ shows that the openset U can be chosen saturated with respect to the orbits of this action.Finally, an argument based on the constancy of the characters of anexterior differential system, similar to that employed in the proof ofthe finiteness theorem below, shows that equality holds, at the level m + µ , in the open set ρ − m ( U ) . In the next section, we provide a globalproof.The property (b) implies the corresponding property for the linearequations R k . Moreover, the property (d) together with (c) implythat L is complete of order k and, consequently, L is a Lie pseudo-algebra (infinitesimal pseudo-group) of order k since J k L (= R k ) is alocally trivial vector sub-bundle of J k T P and J k +1 L is the prolongationof J k L in the sense of linear equations. Finally the property (c), thatwill be a consequence of (a), (b) and the Cartan-Kähler theorem whenthe initial data is real analytic (it will also be a consequence in othersituations, especially in the transitive elliptic case), serves to assure
N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 47 later that any orbit of the infinitesimal action is open in the corre-sponding orbit of the finite action and, more precisely, is a connectedcomponent.We shall say that Γ is transitive when there exists ϕ ∈ Γ , with ϕ ( x ) = y , whatever the points x, y ∈ P and that it is locally trivial when theprojection α × β : J k Γ −→ P × P is a submersion (we do not assumetransitivity ([19]). On account of the property (a), it will suffice to havelocal triviality at order k . We shall say that L is transitive or that Γ is infinitesimally transitive when the vector sub-space induced by L atevery point y in P is equal to T y P . Finally, the formal transitivity isthe one linked to the linear and non-linear equations J k L and J k Γ andcoincides entirely with the transitivity above.Let ( E, π, P, p ) be a finite or infinitesimal prolongation space and Γ ,resp. L , a finite or infinitesimal Lie pseudo-group (pseudo-algebra) oforder k operating on P . The Definition 1 can be re-written by replac-ing the general pseudo-group and pseudo-algebra of all local finite orinfinitesimal transformations by the more specific data Γ and L . Inthis context, we can re-write essentially all of the section 2 by replacing Π (cid:96) + k P and J (cid:96) + k T P by J (cid:96) + k Γ and J (cid:96) + k L as soon as (cid:96) + k ≥ k . Wecan also transcribe the considerations of the previous section where weshall replace Γ (cid:96) + k , resp. L (cid:96) + k , by the prolongations of Γ , resp. L , thegeneral equivalence problem by the restricted one and, in the Theorem2, the groupoid Π (cid:96) + k P by J (cid:96) + k Γ . Inasmuch, we can rewrite the sections3 and 4 in the restricted context though certain parts and especiallythose concerning the morphisms Φ and Ψ require some additional con-siderations. We shall return to this in a later section.Still in a wider context, we can define finite and infinitesimal prolon-gation spaces relative to given finite or infinitesimal pseudo-groups oftransformations. In other terms, the prolongation operations are onlydefined for the elements of the pseudo-group or pseudo-algebra en-visaged. A most relevant example is provided by the Cartan normal prolongation spaces associated to given pseudo-groups and their quo-tient spaces. In replacing the general pseudo-group by a given one wecan still argue as in the previous sections though, of course, we shallnot forget the inequality (cid:96) + k ≥ k .The Lie pseudo-group Γ is said to be analytic when the manifolds J (cid:96) + k Γ , k ≥ k , are analytic sub-groupoids of Π (cid:96) + k P (supposing ofcourse that π : E −→ P is an analytic fibration). Inasmuch, the Lie pseudo-algebra L is said to be analytic when the linear equations J k L are analytic vector sub-bundles of J k T P . Clearly, the analiticity of Γ implies that of L and the converse is also true since the differentiablestructure of J k Γ is entirely determined, in a neighborhood of the unitshence everywhere, by the structure of R k .9. The role of the differential invariants - finitenesstheorems
The interesting situation from the point of view of the differentialinvariants is that of continuous families of orbits. We therefore assume,for the time being, that there exists an integer k such that, for k ≥ k ,the orbits of the action of Γ (cid:96) + k on J k E or rather those of the infinites-imal action of L (cid:96) + k are distributed along a regular foliation i.e. , theintegrable distribution ∆ k is locally of constant dimension. A first inte-gral of ∆ k (a function that is locally constant on each integral leaf of ∆ k or, equivalently, a function whose differential df vanishes on ∆ k ) willbe called a differential invariant of order k of the Lie pseudo-group Γ ,resp. of the pseudo-algebra L , and relative to the prolongation space E . Since ∆ k is assumed to be regular there exists, in a neighborhoodof each point in J k E , a fundamental system of independent differen-tial invariants their number (rank) being equal to the co-dimension of ∆ k . On the other hand, with the aid of the formal derivatives (to-tal derivatives) in the jet manifolds, it is possible to ascend (lift), ina non-trivial way, differential invariants defined on J k E to new differ-ential invariants defined on J k +1 E . Lie’s finiteness theorem for thedifferential invariants states essentially that the invariants of any orderare generated by those of a certain finite order together with all theirsuccessive formal derivatives. The mechanism involving the differentialinvariants presumes of course certain regularity hypotheses as well asspecific technicalities that eventually will lead us to the FundamentalTheorem of Sophus Lie ([24]) and we shall try to describe these in themost succinct manner by referring as much as possible to [20]. Sincethe specific case of prolongation spaces and of the formal equivalenceof structures, our main concern, isn’t but a special case of the generalproblem discussed in the previous reference, it is possible to simplifytwo of the hypotheses and the form under which we state the Lie The-orem for its applications in the equivalence problem. We shall in factprovide a much more precise statement than the one claimed in theTheorem 23.6 ( loc.cit. ). N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 49
Let us first remark that the problem in [20] consists in taking an arbi-trary fibration (surmersion) π : P −→ M together with a sheaf L (Lie sheaf) of vector fields on P and thereafter study the differentialinvariants of L in the realm of the standard prolongation spaces J k P above P . Here, quite to the contrary, we are given an infinitesimal Liepseudo-algebra L of order k on the manifold P , an infinitesimal pro-longation space ( E, π, P, p ) of finite order (cid:96) and study the differentialinvariants of the infinitesimal action of L on the prolongation spaces J k E . We can re-conduce our considerations to the above mentionedcontext by simply considering the prolonged sheaf L (cid:96) = p L defined onthe space E and study the differential invariants of L (cid:96) by the tech-niques and methods found in [20]. However, the present methods arefar more reaching and accurate.We examine initially the hypothesis H ( loc.cit. , pg.363) or, by prefer-ence, the weaker hypothesis on pg.378. H (cid:48) ,Y : There exists an integer k such that, for any k ≥ k , thefibre space ( ˜ L V ) k with base space J k +1 E is of constant rank in theneighborhood of each point Y k +1 .Let us recall (see [20] for the notations) that ( ˜ L V ) k ⊂ J k +1 P × P ˜ J k V E and that this fibre space is the image of ˜ L k = ˜ J k L , L being a Lie sheafover P , by the verticalisation operation described in terms of the exactsequence (22.29) in [20]. However, in the present case , we start withan infinitesimal Lie pseudo-algebra L of order k defined on P , considerits prolongation L (cid:96) = p L to E and ˜ L k becomes ˜ J k ( L (cid:96) ) . Under theseconditions, ( ˜ L V ) k is the image of J k +1 E × P J (cid:96) + k L by the mapping J k +1 E × P J (cid:96) + k T P −→ J k +1 E × P ˜ J k V E defined by ( j k +1 σ ( y ) , j (cid:96) + k ξ ( y )) (cid:55)−→ j k [( pξ ) ◦ σ − ( T σ ◦ ξ )]( y ) . Let us next consider the exact sequence −→ N (cid:96) + k −→ J k +1 E × P J (cid:96) + k L −→ ( ˜ L V ) k −→ The "tilde" notation refers to the composite fibration
T E −→ E −→ P andwhere T E is also replaced by
V E where N (cid:96) + k denotes the kernel. We thus see that the regularity of ( ˜ L V ) k can be replaced, when (cid:96) + k ≥ k , by that of N (cid:96) + k for which thedefining equation is given by j k [( pξ ) ◦ σ − ( T σ ◦ ξ )]( x ) = 0 . This equation can be envisaged as a linear differential equation of order (cid:96) + k in J (cid:96) + k L (or as well in J (cid:96) + k T P ) whose coefficients depend on theparameters in J k +1 E . Consequently, the regularity of this equationin the neighborhood of a jet Z k +1 ∈ J k +1 E , is closely related to thegeometry of the prolongation space E in the neighborhood of β ( Z k +1 ) .As for the other two hypotheses on the pg.363, we can partly weaken H ,X by taking into account that J (cid:96) + k L is a Lie equation hence thedistribution ∆ k automatically satisfies the involutivity condition. How-ever, we shall be forced to strengthen the part concerning regularity.Inasmuch, we shall strengthen the point-wise hypothesis H ,X by a lo-cal condition, its most efficacious verification criterion being providedby the Proposition 25.4 in [20] on account of its Corollary. We thereforeconsider the following hypotheses: H : For any k ≥ k ( ≥ k − (cid:96) ) , the vector bundle N (cid:96) + k has constantrank in a neighborhood of Z k +1 and we denote by k ( Z ) the integerwhere-after (∆ k − ,k ) Z k (the kernel) becomes involutive. H : There exists a family ( U k ) k ≥ k ( Z of open neighborhoods of thejets Z k such that ρ k,k + h : U k + h −→ U k is a fibration and ∆ k hasconstant dimension on U k . H : β ( R k ( Z ) ( L )) Z (cid:48) ) = T β ( Z (cid:48) ) P for all Z (cid:48) ∈ U k ( Z ) and for someinteger k ≥ k .The last hypothesis assures the existence of a local basis of admissible formal derivations of order k centered around Z , admissible meaningthat such derivations transform differential invariants into differentialinvariants. Before stating the desired theorem, let us examine a littlecloser the above hypotheses in order to better discern their meaning.1. Let (∆ k − ,k ) Z (cid:48) be the kernel of T ρ k − ,k : ∆ k −→ ∆ k − at thepoint Z (cid:48) ∈ J k +1 E (this mapping being always surjective). The firsthypothesis serves to prove that there is an order k (cid:48) such that the kernel (∆ k,k +1 ) Z k +1 , k ≥ k (cid:48) , is contained, by means of the canonical identi-fication, in the algebraic prolongation of (∆ k − ,k ) Z k and consequently N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 51 becomes equal to it from an order k (cid:48)(cid:48) onwards or, in other terms, theSpencer δ − complex constructed with the kernels (∆ k − ,k ) Z k becomes − acyclic for k ≥ k (cid:48)(cid:48) . Likewise, it will become involutive beginningwith an integer that we shall denote by k ( Z ) . This hypothesis aloneenables us to prove the asymptotic stability result ([20], Theorems 22.1and 23.1) that in turn and with the aid of the hypotheses H ,Z and H ,Z , leads to the Lie Theorem ( loc.cit. , Theorem 23.6).2. The hypothesis H assures a sufficient number i.e. , a complete setof k − th order differential invariants defined on the open set U k .3. The hypothesis H enables us to obtain a sufficient number of ( k + 1) − st order differential invariants by taking admissible formalderivations of the k − th order differential invariants (and, of course,lifting also the latter up to order k + 1 ).4. One shows that the finiteness property of the differential invariantstakes place at the stage k (cid:32) k + 1 ( i.e. , for the germs of invariants atthe points Z k and Z k +1 respectively) if and only if (∆ k,k +1 ) Z k +1 is thealgebraic prolongation of (∆ k − ,k ) Z k ([20], Lemmas 23.3 and 23.5).We next remark that the three hypotheses, per se independent, are notstrictly necessary to prove the desired results. In fact, the hypothesis H ,X underlying the theorem 23.8 in [20] is considerably weaker than H . However, the asymptotic stability, consequence of H (cid:48) ,Z , joint to H ,Z imply the local regularity and the integrability of the distribution ∆ k , for k > k , in view of the Corollary 5 ( loc.cit. , pg.377, conditions(I) and (II)). Viewed from another angle, we note that solely the hy-potheses H and H will, in virtue of the lemma 23.3 in [20], imply that (∆ k,k +1 ) Z k +1 ⊂ p (∆ k − ,k ) Z k and we thus obtain the asymptotic stabilityof the kernels starting from a certain integer k (cid:48)(cid:48) . These remarks simplyshow that the usage of the above three hypotheses admits a certain flex-ibility, the appropriate choices being conditioned to the results lookedfor.At present we choose H and H as underlying hypotheses and fix theorder µ = k ( Z ) where after the symbols (∆ k − ,k ) Z k become involutive(the hypothesis H only reappearing later when the regularity of the ∆ k becomes apparent). We can further assume that k ( Z ) < k ( Z ) .The hypothesis H implies that the kernels ∆ µ,µ +1 and ∆ µ − ,µ are ofconstant dimension in U µ +1 and U µ respectively, and further (∆ µ − ,µ ) Z µ is involutive, (∆ µ,µ +1 ) Z µ +1 being its algebraic prolongation. According to the Theorem 23.6 ( loc.cit. ), there exists an open neighborhood U µ +1 of Z µ +1 such that (∆ µ,µ +1 ) Z (cid:48) µ +1 = p (∆ µ − ,µ ) Z (cid:48) µ for all Z (cid:48) µ +1 ∈ U µ +1 and consequently the finiteness property of the differential invariantsis verified at the step U µ (cid:32) U µ +1 , U µ = ρ ( U µ +1 ) . Let us nextobserve that the characters τ i of (∆ µ − ,µ ) Z (cid:48) µ , Z (cid:48) µ ∈ U µ , are lowersemi-continuous. The dimensions of ∆ µ − ,µ and ∆ µ,µ +1 being constant,the characterization of the involutivity ([20], §
24, property 8, [23],proposition 6.1) implies the existence of an open neighborhood W µ of Z µ in which the kernels (∆ µ − ,µ ) Z (cid:48) µ , Z (cid:48) µ ∈ W µ are all involutive, thecharacters τ i remaining constant. Let us denote by W µ +1 the inverseimage of W µ with respect to the projection ρ : U µ +1 −→ U µ and,similarly, define W µ + h considering ρ : U µ + h −→ U µ . Furthermore,denote by (∆ (cid:48) µ +2 ) Z (cid:48) the sub-space of T Z (cid:48) J µ +2 E , Z (cid:48) ∈ W µ +2 , definedby (∆ (cid:48) µ +2 ) Z (cid:48) = ker Z (cid:48) { ρ ∗ µ +1 ,µ +2 df, ∂ ϕ df | Z (cid:48) µ ∈ W µ , f ∈ ( I µ +1 ) Z (cid:48)(cid:48) ,ϕ ∈ R µ +1 ( L ) Z (cid:48)(cid:48) , Z (cid:48)(cid:48) = ρ µ +1 ,µ +2 Z (cid:48) } , where I µ +1 denotes the algebra of all differential invariants of order µ + 1 . Since the elements of R µ +1 ( L ) Z (cid:48)(cid:48) are admissible, the inclusion ∆ (cid:48) µ +2 ⊃ ∆ µ +2 holds and the lemma 23.3 in [20] shows furthermore that dim (∆ (cid:48) µ +2 ) Z (cid:48) = dim (∆ µ +1 ) Z (cid:48)(cid:48) + dim p (∆ µ,µ +1 ) Z (cid:48)(cid:48) . However, (∆ µ,µ +1 ) Z (cid:48)(cid:48) , Z (cid:48)(cid:48) ∈ W µ +1 , is involutive it being the prolonga-tion of an involutive space and the property 9 in §
24 of [20] or else, theProposition 9.4 in [23] shows, in view of the constancy of the characters τ i , that dim p (∆ µ,µ +1 ) Z (cid:48)(cid:48) is constant in W µ +1 , the characters of theseprolonged spaces being also constant. Returning to the point Z µ +2 ,we perceive that this dimension is equal to dim (∆ µ +1 ,µ +2 ) Z µ +2 andconsequently that dim (∆ (cid:48) µ +2 ) Z (cid:48) = dim (∆ µ +2 ) Z (cid:48) µ +2 . Furthermore,this entails, in virtue of the constancy of the dimensions of ∆ µ +2 ,that (∆ (cid:48) µ +2 ) Z (cid:48) = (∆ µ +2 ) Z (cid:48) for all Z (cid:48) ∈ W µ +2 . We thus infer that thefiniteness property for the differential invariants is verified at the step W µ +1 (cid:32) W µ +2 . An inductive argument will finally prove, based onthe constancy of the characters, that the finiteness property is verifiedat the stage W µ + h (cid:32) W µ + h +1 since the involutivity as well as theconstancy of the characters is preserved by prolongation.Let us observe that the involutivity property of the kernels (∆ µ − , µ ) Z (cid:48) µ together with the regularity of the ∆ k , k ≥ µ − , on the open sets U k , that "fibrate" one upon the other, serve uniquely to ensure the ex-istence of a family of open neighborhoods W k of Z k , fibering one above N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, I 53 the other in such a way that, along every element Z (cid:48) ∈ proj lim W k ,the consecutive kernels of the ∆ k constitute a − acyclic Spencer δ − com-plex. In the applications, this local − acyclicity property, sole to assurethe finiteness mechanism of the differential invariants, might be veri-fied long before the involutivity. The argument as well as the aimsof the above discussion are somewhat quite the opposite of what hasbeen looked for in the §
24 of [20] where the problem posed was theregularity of the trajectories.
Theorem (of finiteness) 3.
Let ( E, π, P, p ) be an infinitesimal pro-longation space, Z ∈ J ∞ E an infinite jet of a structure of species E and L an infinitesimal pseudo-algebra (Lie pseudo-algebra) operating on P.Assuming that the hypotheses H and H are satisfied at the point Z,we write µ = k and take a family of n (= dim P ) local sections ϕ i of R µ +1 ( L ) , defined in a neighborhood of Z µ +1 , such that { βϕ i ( Z µ +1 ) } generates the tangent space T y P , y = α ( Z ) i.e., the family { ϕ i } isa local basis of admissible formal derivations in the neighborhood of Z µ +1 . Under these conditions, there exists a family ( W k ) k ≥ µ , each W k being an open neighborhood of Z k , such that: i ) ρ k,k + h : W k + h −→ W k is a fibration. ii ) (∆ k,k +1 ) Z (cid:48) k +1 = p (∆ k − ,k ) Z (cid:48) k , Z (cid:48) k +1 ∈ W k +1 . iii ) (∆ k ) Z (cid:48) k = ker Z (cid:48) k { df | f ∈ I } , Z (cid:48) k ∈ W k . iv ) (∆ k +1 ) Z (cid:48) k +1 = ker Z (cid:48) k +1 { ρ ∗ k,k +1 df, ∂ϕ i df | f ∈ I k , ≤ i ≤ n } , Z (cid:48) k +1 ∈ W k +1 . v ) (∆ k +1 ) Z (cid:48) k +1 = ker Z (cid:48) k +1 { ρ ∗ k,k +1 df, ∂ϕ i df | f ∈ I k , ϕ i ∈ R µ +1 ( L ) Z (cid:48) µ +1 } , Z (cid:48) k +1 ∈ W k +1 . vi ) { ρ ∗ k,k +1 df, ∂ϕ i df | f ∈ I k , ≤ i ≤ n } Z (cid:48) k +1 generates ( d I k +1 ) Z (cid:48) k +1 ) , Z (cid:48) k +1 ∈ W k +1 . The interest of the finiteness theorem for the equivalence of structuresis due to the fact that it enables us to translate the equivalence by only a finite number of conditions (equality of the values taken by afinite number of differential invariants). We terminate here this awfullylong "preamble" and will retake the effective study of the equivalenceproblem in part II of this paper where diverse mises en scène shall be examined. As a last word, we should say that all the previousdiscussion can be carried out in the context of prolongation spaces relative to given finite Lie pseudo-groups or infinitesimal Lie pseudo-algebras and also it is worthwhile to recall that Sophus Lie providedsome of the most remarkable contributions. Surprisingly, the formula25.5, concerning the bracket of formal and holonomic derivations ([20])is already written in his work [24] (see also [18], [29]).
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